To find linearly independent solutions of the given third-order differential equation, we can assume a solution of the form y(t) = e^(rt), where r is a constant.
Substituting this into the differential equation, we obtain the characteristic equation:
r^3 + 3r^2 - 29r - 55 = 0
By solving this cubic equation, we can find the values of r that will give us the linearly independent solutions.
Using various methods such as factoring, synthetic division, or numerical approximation, we can find that the roots of the characteristic equation are r = -5, -1, and 11.
To obtain the corresponding linearly independent solutions, we can plug these roots back into the assumed form y(t) = e^(rt). Thus, we have three linearly independent solutions:
y1(t) = e^(-5t)
y2(t) = e^(-t)
y3(t) = e^(11t)
These three solutions form a basis for the solution space of the given differential equation. The general solution can be expressed as a linear combination of these solutions, with arbitrary constants c1, c2, and c3:
y(t) = c1e^(-5t) + c2e^(-t) + c3e^(11t)
This general solution represents all possible solutions to the third-order differential equation, with different choices of the constants providing specific solutions. The linear independence of the solutions ensures that any linear combination of them will not result in redundant or equivalent solutions.
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00 The limit of the sequence {-(; 104 n + e-141 Zn + tan 1(73 n) 6)} is n=1 Hint: Enter the limit as a logarithm of a number (could be a fraction).
The limit of the sequence {-(; 104 n + e-141 Zn + tan 1(73 n) 6)} as n approaches infinity can be summarized as follows: The limit does not exist. The sequence does not converge to a specific value or approach any particular number as n tends to infinity.
To determine the limit of the given sequence, we need to evaluate the terms as n becomes arbitrarily large. Let's break down the sequence: {-(; 104 n + e-141 Zn + tan 1(73 n) 6)}.
The first term, 104n, grows linearly with n. As n approaches infinity, this term also increases without bound.
The second term, e-141Zn, involves the exponential function with a negative exponent. As n tends to infinity, the value of this term approaches zero since any positive base raised to a negative exponent becomes infinitesimally small.
The third term, tan(1(73n)6), involves the tangent function. The argument inside the tangent function, 1(73n)6, increases without bound as n approaches infinity. However, the tangent function oscillates between positive and negative values, and it does not converge to a specific number.
Since the terms in the sequence do not converge to a single value, the limit of the sequence as n approaches infinity does not exist.
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An outdoor amphitheater has 29 seats in the first row, 31 in the second row, 33 in the third row and so on. There are 34 rows together How many can the anghtwater er The amphitheater can seat people
The amphitheater can seat 2108 people.
To find the total number of seats in the amphitheater, we need to sum up the number of seats in each row.
The pattern shows that the number of seats in each row increases by 2 seats compared to the previous row. So, we can create an arithmetic sequence to represent the number of seats in each row.
The first term (a) is 29, the common difference (d) is 2, and the number of terms (n) is 34.
Using the formula for the sum of an arithmetic series:
Sn = (n/2)(2a + (n-1)d)
Plugging in the values:
S34 = (34/2)(2(29) + (34-1)(2))
S34 = 17(58 + 66)
S34 = 17(124)
S34 = 2108
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Consider an infinitely repeated game in which, in each period, two firms with zero costs choose quantities and prices are given by: Pi = 1 -q1-q2/2, P2 = 1 - q2-q 1/2. Firms have a common discount factor of d = 1/2. a) Explain what a trigger strategy is and determine whether the firms can attain the joint profit maximising outcome in a subgame perfect equilibrium using trigger strategies. b) Explain what a stick and carrot strategy is and discuss whether it is possible to attain the joint-profit maximising outcome in a subgame perfect equilibrium using stick and carrot strategies.
A trigger strategy is a strategy that specifies an action to take in response to certain observed actions by other players. In this context, a trigger strategy involves cooperating as long as the other player cooperates, but immediately defecting and pursuing a different strategy if the other player deviates from cooperation.
In the given game, the firms cannot attain the joint profit-maximizing outcome in a subgame perfect equilibrium using trigger strategies because there is no trigger that can effectively sustain cooperation in the repeated game. Both firms have an incentive to deviate and lower their price to increase their own profit.
A stick and carrot strategy combines punishment for deviating from cooperation (stick) and rewards for cooperating (carrot). In this case, a stick and carrot strategy could involve punishing the deviating firm by setting a low quantity or price in response to their deviation, while rewarding cooperation by maintaining high quantities and prices. However, it is unlikely to attain the joint-profit maximizing outcome in a subgame perfect equilibrium using stick and carrot strategies because the firms still have an incentive to deviate and lower their price to increase their own profit, even if they face punishments or rewards. Therefore, sustaining cooperation and achieving the joint-profit maximizing outcome is challenging in this repeated game.
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It can be shown that if events are occurring in time according to a Poisson distribution with mean
λt
then the interarrival times between events have an exponential distribution with mean 1/λ
(a)Suppose that customers arrive at a checkout counter at the rate of two per minute.
What are the mean (in minutes) and variance of the waiting times between successive customer arrivals?
mean = min
variance =
(b)
If a clerk takes 3.2 minutes to serve the first customer arriving at the counter, what is the probability that at least one more customer will be waiting when the service to the first customer is completed? (Round your answer to four decimal places.)
The time it takes to serve each customer in a queue is one way to measure waiting times in queueing theory. According to the Poisson distribution, if events are happening in time, the probability that exactly k events occur in a given time period is given by:P(k,λ) = (λ^k * e^(-λ))/k!where λ is the average number of events per unit time, and k! denotes k factorial, which is the product of all positive integers up to k.
Here, we're looking at the probability of there being at least one customer in line when the first customer is finished being served. The inter-arrival time is exponential, with a mean of 3.2 minutes. This means that the rate at which customers arrive is λ = 1/3.2 per minute.
Using the Poisson distribution, the probability that at least one customer is in line when the first customer is finished is:P(at least 1 customer in line) = 1 - P(0 customers in line) = 1 - P(0,λ')where λ' is the rate at which customers arrive during the time it takes to serve the first customer.
Since this time is 3.2 minutes, λ' = λ * 3.2 = 1.0.P(0,1.0) = (1.0^0 * e^(-1.0))/0! = 0.3679P(at least 1 customer in line) = 1 - P(0,1.0) = 1 - 0.3679 = 0.6321The probability that at least one more customer will be waiting when the service to the first customer is completed is 0.6321 (rounded to four places).
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Find the equilibrium point. Then find the consumer and producer surplus. 14) D(x) = - 3x + 6, S(x) = 3x + 2
To find the equilibrium point, we need to set the demand (D(x)) equal to the supply (S(x)):
[tex]-3x + 6 = 3x + 2[/tex]
Now, let's solve for x:
[tex]-3x - 3x = 2 - 6[/tex]
[tex]-6x = -4\\x = (-4) / (-6)\\x = 2/3[/tex]
Therefore, the equilibrium point is x = 2/3.
To find the consumer surplus, we need to calculate the area between the demand curve and the equilibrium quantity.
Consumer Surplus = [tex]∫[0, 2/3] (D(x) - P) dx[/tex]
Since the price (P) is not given, we cannot calculate the exact consumer surplus without additional information.
Similarly, to find the producer surplus, we need to calculate the area between the supply curve and the equilibrium quantity.
Producer Surplus =[tex]∫[0, 2/3] (P - S(x)) dx[/tex]
Without knowing the price (P), we cannot calculate the exact producer surplus either.
Please provide the price (P) in order to calculate the consumer and producer surplus accurately.
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Can you explain how to solve this problem?
(Please write in neat text to read clearly)
7.21 The number of customers, K, that shop at the neighborhood store in a day has the PMF Pk (k) ke k=0,1,2,... k! Independently of K, the number of items N that each customer purchases has the PMF n=
Once I have the PMF for N, I can explain how to use these PMFs to calculate various probabilities or expected values related to the number of customers and items purchased at the neighborhood store.
Let's break down the problem step by step.
The problem states that the number of customers, K, that shop at the neighborhood store in a day follows a probability mass function (PMF) given by Pk(k) = ke^(-k!) for k = 0, 1, 2, ...
We are also given that the number of items, N, that each customer purchases has its own PMF, which is not specified in your question. To solve the problem completely, we need the PMF for N as well. Please provide the PMF for N so that I can proceed with the solution.
Once I have the PMF for N, I can explain how to use these PMFs to calculate various probabilities or expected values related to the number of customers and items purchased at the neighborhood store.
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pThe current price of a non-dividend-paying stock is $73.07 and you expect the stock price to either go up by a factor of 1.105 or down by a factor of 0.914 each period for 2 periods over the next 0.2 years. Each period is 0.1 years long. A European call option on the stock expires in 0.2 years. Its strike price is $73. The risk-free rate is 5% (annual, continuously compounded).
The value of the European call option on the stock, based on the given parameters, is $2.65. This is calculated using the Black-Scholes option pricing model with the provided stock price, strike price, risk-free rate, time to expiration, and implied volatility.
To calculate the value of the European call option, we can use the Black-Scholes option pricing model. The formula for the call option value is:
C = S * N(d1) - X * e^(-rT) * N(d2)
Where:
C is the call option value,
S is the current stock price,
N() represents the cumulative standard normal distribution function,
d1 = (ln(S/X) + (r + σ^2/2) * T) / (σ * √T),
d2 = d1 - σ * √T,
X is the strike price of the option,
r is the risk-free rate,
T is the time to expiration in years, and
σ is the volatility of the stock price.
Plugging in the given values, we have:
S = $73.07,
X = $73,
r = 0.05,
T = 0.2,
σ is the implied volatility.
By calculating the values of d1 and d2 using the provided formula, we can then use the cumulative standard normal distribution function to find N(d1) and N(d2). Finally, substituting all the values into the option pricing formula, we obtain the value of the European call option as $2.65.
This calculation assumes that the stock price follows a lognormal distribution, the market is efficient, and there are no transaction costs or taxes.
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use taylor's formula for at the origin to find quadratic and cubic approximations of f(x,y)=2xe^(2y) near the origin.
The quadratic approximation was found to be f(x, y) ≈ 4xy, while the cubic approximation was f(x, y) ≈ f(0, 0) + 4xy + 4e²ˣxy.
To find the quadratic and cubic approximations of f(x, y), we'll start by finding the first and second partial derivatives of the function at the origin. Then, we'll use these derivatives to construct the polynomial approximations using Taylor's formula.
The partial derivative of f(x, y) with respect to x, denoted as fₐ, can be found by treating y as a constant and differentiating f(x, y) with respect to x: fₐ = ∂f/∂x = 2e²ˣ
Similarly, the partial derivative of f(x, y) with respect to y, denoted as fₓ, can be found by treating x as a constant and differentiating f(x, y) with respect to y: fₓ = ∂f/∂y = 4xe²ˣ
Now, let's find the second partial derivatives:
The second partial derivative of f(x, y) with respect to x, denoted as fₐx, can be found by differentiating fₐ with respect to x: fₐx = ∂²f/∂x² = 0 (since the derivative of 2e²ˣ with respect to x is 0)
Similarly, the second partial derivative of f(x, y) with respect to y, denoted as fₓy, can be found by differentiating fₓ with respect to y: fₓy = ∂²f/∂y² = 8xe²ˣ
The mixed partial derivative of f(x, y) with respect to x and y, denoted as fₐy, can be found by differentiating fₐ with respect to y or fₓ with respect to x: fₐy = ∂²f/∂x∂y = 8e²ˣ
The quadratic approximation involves the first partial derivatives and the second partial derivatives:
f(x, y) ≈ f(0, 0) + fₐ(0, 0)x + fₓ(0, 0)y + (1/2)fₐx(0, 0)x² + (1/2)fₓy(0, 0)y² + fₐy(0, 0)xy
Since we are approximating near the origin (x = 0, y = 0), we substitute these values into the formula:
f(x, y) ≈ f(0, 0) + fₐ(0, 0)x + fₓ(0, 0)y + (1/2)fₐx(0, 0)x² + (1/2)fₓy(0, 0)y² + fₐy(0, 0)xy
Substituting the derivative values we calculated earlier:
f(x, y) ≈ f(0, 0) + 0 + 0 + (1/2) * 0 * x² + (1/2) * 8xe⁰ * y² + 8e⁰ * x * y
Simplifying further:
f(x, y) ≈ f(0, 0) + 4xy
So, the quadratic approximation of f(x, y) near the origin is f(x, y) ≈ 4xy.
The cubic approximation involves the first partial derivatives and the second partial derivatives:
f(x, y) ≈ f(0, 0) + fₐ(0, 0)x + fₓ(0, 0)y + (1/2)fₐx(0, 0)x² + (1/2)fₓy(0, 0)y² + fₐy(0, 0)xy + (1/6)fₐxx(0, 0)x³ + (1/6)fₓyy(0, 0)y³ + (1/2)fₐxy(0, 0)x²y + (1/2)fₐyy(0, 0)xy²
Since the second partial derivative fₐx(0, 0) is zero, and fₐxx(0, 0) is also zero, the cubic approximation simplifies to:
f(x, y) ≈ f(0, 0) + fₐ(0, 0)x + fₓ(0, 0)y + (1/2)fₓy(0, 0)y² + fₐy(0, 0)xy + (1/6)fₓyy(0, 0)y³ + (1/2)fₐyy(0, 0)xy²
Substituting the derivative values we calculated earlier:
f(x, y) ≈ f(0, 0) + 0 + 0 + (1/2) * 8xe⁰ * y² + 8e⁰ * x * y + (1/6) * 0 * y³ + (1/2) * 0 * xy²
Simplifying further:
f(x, y) ≈ f(0, 0) + 4xy + 4e²ˣxy
So, the cubic approximation of f(x, y) near the origin is f(x, y) ≈ f(0, 0) + 4xy + 4e²ˣxy.
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Answer the following question and show all the workings clearly. Submit your answer in pdf file.
Name:.....
ID:
The density of a piece of triangular metal R = {(x, y): 0 ≤x≤1, 0 ≤ y ≤ 2x) is given by the function g(x, y) =5x+5y +5.
Identify the metal piece's centre of mass.
The y-cοοrdinate οf the center οf mass is 31/6.
The center οf mass οf the triangular metal piece is lοcated at (13/12, 31/6).
What is Mass?Mass is a measure οf the amοunt οf matter in a substance οr οbject. The base SI unit fοr mass is the kilοgram (kg), but smaller masses can be measured in grams (g). Yοu wοuld use a scale tο measure weight. Mass is a measure οf the amοunt οf matter an οbject cοntains.
Tο find the center οf mass οf the triangular metal piece, we need tο calculate the cοοrdinates (x, y). The center οf mass cοοrdinates can be determined using the fοllοwing fοrmulas:
x = (1/A) ∫∫x * g(x, y) dA
y = (1/A) ∫∫y * g(x, y) dA
where A is the area οf the triangular metal piece.
First, let's find the area οf the triangular regiοn R:
A = ∫∫R dA
Since the triangular regiοn R is defined as 0 ≤ x ≤ 1 and 0 ≤ y ≤ 2x, the limits οf integratiοn fοr x and y are as fοllοws:
0 ≤ x ≤ 1
0 ≤ y ≤ 2x
Therefοre, the area A can be calculated as:
A = ∫∫R dA = ∫0¹ ∫[tex]0^{(2x)[/tex] dy dx
Integrating with respect tο y first:
A = ∫0¹ (2x - 0) dx = ∫0¹ 2x dx = [[tex]x^2[/tex]]0¹ = 1
The area οf the triangular regiοn R is 1.
Nοw, let's find x:
x = (1/A) ∫∫x * g(x, y) dA
= (1/1) ∫∫R x * (5x + 5y + 5) dA
= 5 ∫∫R [tex]x^2[/tex] + xy + x dA
Integrating with respect tο y first:
x = 5 ∫0¹ ∫[tex]0^{(2x)} (x^2 + xy + x)[/tex] dy dx
= 5 ∫0¹ [[tex](x^2y + (xy^2)/2 + xy)]0^{(2x)[/tex] dx
= 5 ∫0¹ [[tex](2x^3 + (2x^3)/2 + 2x^2)[/tex] - (0 + 0 + 0)] dx
= 5 ∫0¹[tex](3x^3 + x^2)[/tex] dx
= [tex]5 [(3/4)x^4 + (1/3)x^3][/tex]0¹
= 5 [(3/4) + (1/3)]
= 5 [(9/12) + (4/12)]
= 5 (13/12)
= 13/12
Therefοre, the x-cοοrdinate οf the center οf mass is 13/12.
Next, let's find y:
y = (1/A) ∫∫y * g(x, y) dA
= (1/1) ∫∫R y * (5x + 5y + 5) dA
= 5 ∫∫R xy + [tex]y^2[/tex] + 5y dA
Integrating with respect tο y first:
y = 5 ∫[tex]0^1[/tex] ∫[tex]0^{(2x)} (xy + y^2 + 5y)[/tex] dy dx
= 5 ∫[tex]0^1 [(x/2)y^2 + (y^3)/3 + (5/2)y^2]0^{(2x)[/tex] dx
= 5 ∫[tex]0^1 [(x/2)(4x^2) + (8x^3)/3 + (5/2)(4x^2)[/tex]] dx
= 5 ∫[tex]0^1 (2x^3 + (8/3)x^3 + 10x^2)[/tex]dx
= 5 [[tex](1/2)x^4 + (4/3)x^4 + (10/3)x^3]0^1[/tex]
= 5 [(1/2) + (4/3) + (10/3)]
= 5 [(3/6) + (8/6) + (20/6)]
= 5 (31/6)
= 31/6
Therefοre, the y-cοοrdinate οf the center οf mass is 31/6.
The center οf mass οf the triangular metal piece is lοcated at (13/12, 31/6).
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Consider the problem min 22 – (x1 – 2)3 + 3 subject to X2 > 1 Which is the value of u*?
The problem involves minimizing the expression 22 - (x1 - 2)^3 + 3 subject to the constraint X2 > 1. To find the optimal solution, we need to determine the value of u* that satisfies the problem's conditions.
To solve the optimization problem, we can useuse the method of Lagrange multipliers. First, we set up the Lagrangian function L(x1, x2, u) as L = 22 - (x1 - 2)^3 + 3 - u(x2 - 1), where u is the Lagrange multiplier associated with the constraint X2 > 1.
Next, we find the partial derivatives of L with respect to x1, x2, and u and set them equal to zero to find the critical points. Differentiating L with respect to x1 yields -3(x1 - 2)^2 = 0, which gives x1 = 2 as the critical point. Differentiating L with respect to x2 gives -u = 0, leading to u = 0. Finally, differentiating L with respect to u gives x2 - 1 = 0, resulting in x2 = 1.
Since x1 = 2 and x2 = 1 satisfy both the original expression and the constraint, the optimal solution is obtained when u* = 0.
In conclusion, the value of u* for the given optimization problem is 0. This value satisfies the conditions and ensures the optimal solution. The method of Lagrange multipliers is employed to find the critical points and determine the value of u*
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solve the following equations and check your answers: a) log (x+1) - log (x-1)=2 b) 7^x/2 = 5^-1x
a) The solution to the equation log(x+1) - log(x-1) = 2 is x = 3. The check can be done by substituting x = 3 into the original equation and verifying that both sides are equal.
a) To solve the equation log(x+1) - log(x-1) = 2, we can use the properties of logarithms. First, we can simplify the equation using the quotient rule of logarithms:
log((x+1)/(x-1)) = 2
Next, we can rewrite the equation in exponential form:
10^2 = (x+1)/(x-1)
Simplifying further, we have:
100(x-1) = x+1
Distributing and combining like terms:
100x - 100 = x + 1
Subtracting x from both sides and adding 100 to both sides:
99x = 101
Dividing both sides by 99:
x = 101/99
Now, to check our solution, we substitute x = 101/99 back into the original equation:
log((101/99)+1) - log((101/99)-1) = 2
log(200/99) - log(2/99) = 2
Applying the properties of logarithms:
log((200/99)/(2/99)) = 2
Simplifying:
log(100) = 2
This is true since log(100) = 2. Therefore, the solution x = 101/99 satisfies the original equation.
b) The solution to the equation 7^(x/2) = 5^(-x) is x = 0. The check can be done by substituting x = 0 into the original equation and verifying that both sides are equal.
Explanation:
b) To solve the equation 7^(x/2) = 5^(-x), we can take the logarithm of both sides. We can choose any logarithm base, but let's use the natural logarithm (ln) for this explanation:
ln(7^(x/2)) = ln(5^(-x))
Using the logarithm property, we can bring down the exponent:
(x/2)ln(7) = -x ln(5)
Now, we can simplify the equation by dividing both sides by ln(7) and multiplying both sides by 2:
x = -2x ln(5)/ln(7)
We can simplify the right side further by dividing both sides by x:
1 = -2 ln(5)/ln(7)
Now, we can solve for ln(5)/ln(7) by dividing both sides by -2:
-1/2 = ln(5)/ln(7)
Finally, we can solve for ln(5)/ln(7) using the properties of logarithms and exponential form:
e^(-1/2) = 5/7
This means that ln(5)/ln(7) is approximately equal to -1/2. Therefore, substituting x = 0 back into the original equation:
7^(0/2) = 5^(-0)
1 = 1
Both sides are equal, confirming that x = 0 is the solution to the equation.
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Consider the function f(z) = { e^(-1/x^2), z≠0 0, z=0. }
Expand f in a Laurent series.
To expand the function f(z) = { e^(-1/x^2), z≠0; 0, z=0 } in a Laurent series, we need to find the coefficients of the series representation. First, let's rewrite f(z) in terms of z:
f(z) = e^(-1/z^2) for z≠0, and f(z) = 0 for z=0.
Now, let's use the Maclaurin series expansion of e^x:
e^x = Σ (x^n)/n! for n = 0, 1, 2, ...
Replace x with -1/z^2:
f(z) = Σ (-1/z^2)^n / n! for z≠0
Simplify and rewrite it as a Laurent series:
f(z) = Σ (-1)^n / (z^(2n) * n!) for z≠0 and n = 0, 1, 2, ...
This is the Laurent series expansion of the given function f(z).
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thomas invests $105 in an account that pays 5 percent simple interest. how much money will thomas have at the end of 5 years?
Thomas will have $131.25 in his account at the end of 5 years, considering a simple interest rate of 5 percent on his initial investment of $105.
Simple interest is calculated based on the initial amount of money invested, known as the principal, and the interest rate. The formula for calculating simple interest is:
Interest = Principal × Rate × Time
Where:
Principal is the initial amount of money invested.
Rate is the interest rate, expressed as a decimal.
Time is the duration of the investment in years.
In this case, Thomas has invested $105, and the interest rate is 5 percent, which can be written as 0.05 in decimal form. The time period is 5 years. Let's substitute these values into the formula to calculate the interest earned:
Interest = $105 × 0.05 × 5
= $26.25
The interest earned over 5 years is $26.25. To determine the total amount of money Thomas will have at the end of 5 years, we need to add the interest to the initial investment:
Total amount = Principal + Interest
= $105 + $26.25
= $131.25
Therefore, at the end of 5 years, Thomas will have a total of $131.25 in his account.
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The function F(C) =
9/5C + 32
converts a temperature from C degrees Celsius to F degrees
Fahrenheit. (a) Express the temperature in degrees Celsius C as a function
of the temperature in degrees Fahrenheit F.
(b) Verify that C = C(F) is the inverse of F = F(C) by
showing that C(F(C)) = C and F(C(F) ) = F. (c) What is the temperature in degrees Celsius if it
is 70 degrees Fahrenheit?
(a) To express the temperature in degrees Celsius C as a function of the temperature in degrees Fahrenheit F, we need to rearrange the equation F(C) = (9/5)C + 32 to solve for C.
F(C) = (9/5)C + 32
Subtract 32 from both sides:
F(C) - 32 = (9/5)C
Multiply both sides by (5/9):
(5/9)(F(C) - 32) = C
So, the function to convert temperature from Fahrenheit to Celsius is:
C(F) = (5/9)(F - 32)
(b) To verify that C = C(F) is the inverse of F = F(C), we substitute C(F) into F(C) and F(C(F)) into C(F) and check if we get the original values:
F(C(F)) = F[(5/9)(F - 32)]
= (9/5)[(5/9)(F - 32)] + 32
= F - 32 + 32
= F
C(F(C)) = C[(9/5)C + 32]
= (5/9)[(9/5)C + 32 - 32]
= C
Since both C(F(C)) and F(C(F)) result in the original values C and F, we can conclude that C = C(F) is the inverse of F = F(C).
(c) If it is 70 degrees Fahrenheit, we can use the function C(F) = (5/9)(F - 32) to find the temperature in degrees Celsius:
C(70) = (5/9)(70 - 32)
= (5/9)(38)
≈ 21.11 degrees Celsius
Therefore, if it is 70 degrees Fahrenheit, the temperature in degrees Celsius would be approximately 21.11 degrees Celsius.
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2. (20 points) (Order Statistics as Maximum Likelihood Estimates) Suppose Y1, Y2, , Yn is a set of measurements representing an exponential pdf with lambda = 1 but with an unknown "threshold" parameter, θ. That is, fy(y; θ) = e^-(y-θ), y >= θ; θ > 0 - = Find the maximum likelihood estimate for θ.
The maximum likelihood estimate for the threshold parameter θ is the smallest measurement Y1 in the set of measurements. This makes intuitive sense, as the exponential distribution with a threshold parameter θ is simply the exponential distribution shifted to the right by θ units. The smallest measurement in the set represents the point at which the distribution starts, so it is a natural choice for the threshold parameter.
To find the maximum likelihood estimate for θ, we first need to find the likelihood function for the given set of measurements. The likelihood function is the product of the individual probabilities of obtaining each measurement given the value of θ.
Let's assume that the measurements are sorted in ascending order, so that Y1 ≤ Y2 ≤ ... ≤ Yn. Then, the likelihood function is given by:
L(θ) = ∏(i=1 to n) e^-(Yi-θ)
= e^(-Σ(i=1 to n) (Yi-θ))
= e^(-nθ + Σ(i=1 to n) Yi)
Now, to find the maximum likelihood estimate for θ, we need to maximize the likelihood function with respect to θ. We can do this by taking the derivative of the likelihood function with respect to θ and setting it to zero:
d/dθ L(θ) = ne^(-nθ + Σ(i=1 to n) Yi) - ∑(i=1 to n) e^-(Yi-θ)
= 0
Simplifying this equation, we get:
n = ∑(i=1 to n) e^-(Yi-θ)
Taking the natural logarithm of both sides and solving for θ, we get:
θ = Y1
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write a mathematical equation to justify the statement ln(17)=2.833
To justify the statement ln(17) = 2.833 mathematically, we can use the definition of the natural logarithm function.
The natural logarithm of a number x, denoted as ln(x), is defined as the exponent to which the base e (approximately 2.71828) must be raised to obtain the number x.
In this case, we have ln(17) = 2.833. To justify this statement mathematically, we can rewrite it using the definition of the natural logarithm:
e^(2.833) = 17
Here, e represents the base of the natural logarithm function, which is approximately 2.71828. By raising e to the power of 2.833, we should obtain the value of 17.
So, the mathematical equation to justify the statement ln(17) = 2.833 is e^(2.833) = 17.
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Question 16 Not yet answered Points out of 1.00 Flag question Question 17 Not yet answered Points out of 1.00 Flag question Suppose you roll a purple die where each face represents a number from 1 to
The probability of rolling a 3 or a 5 on the purple die is 1/3.
The problem states that we are rolling a purple die, which has six faces representing the numbers 1 to 6. We want to determine the probability of getting a 3 or a 5.
To find the probability, we need to compare the number of favorable outcomes (rolling a 3 or a 5) to the total number of possible outcomes.
The total number of possible outcomes is 6 since there are six faces on the die.
Now let's consider the favorable outcomes. In this case, we are interested in rolling a 3 or a 5. There are two faces on the die that represent these numbers.
Therefore, the number of favorable outcomes is 2.
To calculate the probability, we divide the number of favorable outcomes by the total number of possible outcomes:
Probability = Number of Favorable Outcomes / Total Number of Possible Outcomes
In this case, the probability is 2/6.
This fraction can be simplified by dividing both the numerator and denominator by their greatest common divisor, which is 2:
2/6 = 1/3
Therefore, the probability of rolling a 3 or a 5 on the purple die is 1/3.
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Incomplete question:
Suppose you roll a purple die where each face represents a number from 1 to 6. Determine the probability of getting a 3 or a 5.
The vector field F with rightwards arrow on top left parenthesis x comma y right parenthesis equals open angle brackets s e c squared x comma space 3 y squared close angle brackets is conservative.
Find f left parenthesis x comma y right parenthesis such that F with rightwards arrow on top equals nabla f .
a.
f equals 2 space s e c x plus 6 y
b.
f equals y tan x plus x y cubed
c.
f equals y cubed minus tan x
d.
f equals fraction numerator y cubed tan x over denominator 3 end fraction
e.
f equals tan x plus y cubed
The potential function for the vector field[tex]F = (sec^{2} x, 3y^{2})[/tex] is f(x, y) = [tex]tan(x) + y^{3}[/tex].
To determine the potential function f such that the vector field is [tex]F = (sec^{2} x, 3y^{2})[/tex]conservative, we need to find f(x, y) that satisfies the condition ∇f = F.
Taking the partial derivatives of the potential function f(x, y) with respect to x and y, we get:
[tex]\partial f/\partial x = sec^{2}x[/tex]
[tex]\partial f/\partial y = 3y^{2}[/tex]
To find f(x, y), we integrate each partial derivative with respect to its respective variable:
[tex]\int\limits sec^{2}x dx = tan x + C(y)[/tex]
[tex]\int\limits 3y^{2} dy = y^{3} + C(x)[/tex]
Since f(x, y) is a potential function, it should be independent of the variable we integrate with respect to. Therefore, C(x) and C(y) must be constant functions.
From the above integrals, we obtain:
[tex]f(x, y) = tan x + C(y) = y^{3} + C(x)[/tex]
To find the potential function, we equate the constant functions:
[tex]C(y) = y^{3} + C(x)[/tex]
This equation implies that the constant functions C(y) and C(x) must be equal to the same constant value, let's call it C.
Therefore, the potential function f(x, y) is given by:
[tex]f(x, y) = tan x + y^{3}+ C[/tex]
Now, comparing this potential function with the given options, we find that option (e) is the correct answer:
[tex]f(x, y) = tan x + y^{3}[/tex]
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for which of the following correlations would the data points be clustered most closely around a straight line?
The closer the correlation coefficient is to +1, the more closely the data points will be clustered around the straight line.
The correlation for which the data points would be clustered most closely around a straight line is a strong positive correlation. In this type of correlation, as one variable increases, the other variable also increases at a consistent rate, resulting in a straight line when the data points are plotted. The closer the correlation coefficient is to +1, the more closely the data points will be clustered around the straight line.
For the following correlations, the data points would be clustered most closely around a straight line when the correlation coefficient is closest to 1 or -1. A positive correlation near 1 indicates a strong positive relationship, while a negative correlation near -1 indicates a strong negative relationship. In both cases, the data points will be tightly clustered around a straight line.
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a correlation coefficient of +0.95 or -0.95 would result in the data points being clustered most closely around a straight line.
The strength and direction of the correlation determine how closely the data points cluster around a straight line. In general, a stronger correlation indicates that the data points are more closely clustered around a straight line.
Therefore, for the following correlations, the data points would be clustered most closely around a straight line in the case of a correlation coefficient of +0.95 or -0.95. These correlation coefficients indicate a strong positive or negative linear relationship between the variables, respectively. The data points would be tightly clustered around a straight line with little scatter, indicating a high degree of linear association between the variables.
Correlation coefficients of +0.70, -0.70, and 0.10 indicate moderate positive, moderate negative, and weak positive correlation, respectively. While these correlations also show some degree of clustering around a straight line, it would not be as tight and pronounced as with correlation coefficients of +0.95 or -0.95.
In summary, a correlation coefficient of +0.95 or -0.95 would result in the data points being clustered most closely around a straight line.
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help please
y The growth of circular colonies of bacteria is given by dy 0.6y, where t is the number of hours and y is dt the number of bacteria in thousands. Assume that there were 36 thousand bacteria initially
The growth of the colony of bacteria is given by the function y = 36e^(0.6t), where t is the time in hours and y is the number of bacteria in thousands.
The given differential equation is:
dy/dt = 0.6y
We can solve this using separation of variables by writing it in the form:
1/y dy = 0.6 dt
Integrating both sides, we get:
ln|y| = 0.6t + C
where C is the constant of integration.
To find the value of C, we use the initial condition that there were 36 thousand bacteria initially:
y(0) = 36
Substituting t = 0 and y = 36 into the above equation, we get:
ln|36| = 0.6(0) + C
C = ln|36|
So the solution to the differential equation is:
ln|y| = 0.6t + ln|36|
Simplifying, we get:
ln|y| = ln|36| + 0.6t
Taking the exponential of both sides, we get:
|y| = e^(ln|36|+0.6t)
Simplifying further, we get:
y = ± 36e^(0.6t)
Since the number of function cannot be negative, we take the positive solution:
y = 36e^(0.6t)
Therefore, the growth of the colony of bacteria is given by the function y = 36e^(0.6t), where t is the time in hours and y is the number of bacteria in thousands.
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find the slope of the tangent line to y=x^2-1 y = x 2 − 1 at p=(1,0) p = ( 1 , 0 ) .
The slope of the tangent line to y=x^2-1 y = x 2 − 1 at p=(1,0) p = ( 1 , 0 ) is 2. The derivative of y=x^2-1 y = x 2 − 1 is 2x 2 x , so the slope of the tangent line at x=1 x = 1 is 2(1) = 2.
The slope of the tangent line to y=x^2-1 y = x 2 − 1 at p=(1,0) p = ( 1 , 0 ) , we need to take the derivative of the function y=x^2-1 y = x 2 − 1 and evaluate it at x=1 x = 1 , which will give us the slope of the tangent line at p=(1,0) p = ( 1 , 0 ) .The slope of the tangent line to y=x^2-1 y = x 2 − 1 at p=(1,0) p = ( 1 , 0 ) is 2. The slope of the tangent line to y=x^2-1 at the point P=(1,0). To find the slope, we'll need to use the derivative of the function, which represents the instantaneous rate of change.
The function we are working with is y=x^2-1. To find its derivative, we can use the power rule: dy/dx = 2x. Now, we have the general formula for the slope of the tangent line at any point on the curve. At the specific point P=(1,0), we can substitute x=1 into the derivative formula to find the slope of the tangent line: dy/dx = 2(1) = 2. So, the slope of the tangent line to y=x^2-1 at P=(1,0) is 2.
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This exercise involves the formula for the area of a circular sector The area of a sector of a circle with a central angle of Arad i 20 m. Find the rol of the circle Cound your answer to decimal place
To find the radius of a circle given the area of a sector and the central angle, we can use the formula for the area of a sector:
Area = (θ/360) * π * r²,
where θ is the central angle in degrees, π is the mathematical constant pi (approximately 3.14159), and r is the radius of the circle.
In this exercise, we are given the area of the sector as 20 square meters. Let's assume the central angle is A degrees. Plugging in the values, we have:
20 = (A/360) * π * r².
To find the radius r, we rearrange the equation:
r² = (20 * 360) / (A * π).
Taking the square root of both sides, we get:
r = √[(20 * 360) / (A * π)].
Calculating the expression inside the square root and substituting the given central angle A, we can find the value of r to the desired decimal place.
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A2. Let X., X2,..., Xu be av.s of size from a gamma distribution with shape parameter x = 4 t rate parameter B=0. X ~ Gamma (4,0) a) find the fisher information b) Show that the MLE of o is efficient for o. c) Find the 95% confidence interval for o using the lim limiting property of MLE'S
The Fisher information for given gamma distribution with α = 4 and β = 0 can be calculated. The MLE of β is shown to be efficient for β and 95% confidence interval is determined using the limiting property of MLEs.
(a) The Fisher information measures the amount of information that a random sample carries about an unknown parameter. For the given gamma distribution with shape parameter α = 4 and rate parameter β = 0, the Fisher information can be calculated as I(β) = [tex]\frac{n}{\beta ^{2} }[/tex], where n is the sample size.
(b) To show that the MLE of the rate parameter β is efficient for β, we need to demonstrate that it achieves the Cramér-Rao lower bound, which states that the variance of any unbiased estimator is greater than or equal to the reciprocal of the Fisher information. Since the MLE is asymptotically unbiased and achieves the Cramér-Rao lower bound, it is efficient.
(c) Using the limiting property of MLEs, we can construct a confidence interval for β. As the sample size increases, the MLE follows an approximately normal distribution. The 95% confidence interval can be calculated as [tex]\beta[/tex] ± [tex]1.96(\frac{1}{\sqrt{I(\beta )} } )[/tex], where [tex]\beta[/tex] is the MLE estimate and I(β) is the Fisher information.
By substituting the values of α and β into the formulas we can obtain the specific results for this gamma distribution.
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Find the area of the region that lies inside the first curve and outside the second curve. r = 10 cos 0, r = s Find the area of the region that lies inside the first curve and outside the second curve.
r = 7 cos , r = 3 + cos 0 Find the area of the region that lies inside both curves.
r= √3 cos 0, r = sin 0
4/9
The area of the region between the curves r = √3 cos(θ) and r = sin(θ) is approximately 0.478 square units.
To find the area of the region that lies inside the first curve and outside the second curve, we need to evaluate the integral of the difference between the two curves with respect to the angle.
For the curves r = 10 cos(θ) and r = θ, we can find the points of intersection by setting the two equations equal to each other:
10 cos(θ) = θ
This equation cannot be solved analytically, so we need to approximate the points of intersection numerically. One of the points of intersection is near θ ≈ 0.739, and the other point is near θ ≈ 4.493.
To calculate the area between the curves, we integrate the difference of the curves from the smaller angle to the larger angle:
A = ∫[tex](r_{1} ^2 - r_{2} ^2)[/tex]dθ
Where r₁ is the larger curve (r = 10 cos(θ)) and r₂ is the smaller curve (r = θ).
A = ∫((10 cos(θ))² - θ²) dθ
Integrating this expression over the range of θ from the smaller point of intersection to the larger point of intersection will give us the area of the region.
A = ∫[100 cos²(θ) - θ²] dθ
Evaluating this integral analytically is challenging, so we'll need to approximate it numerically or use numerical integration methods.
Using numerical integration techniques or software, we find that the area of the region between the curves r = 10 cos(θ) and r = θ is approximately 32.076 square units.
For the second problem with the curves r = 7 cos(θ) and r = 3 + cos(θ), we need to find the points of intersection by equating the two equations:
7 cos(θ) = 3 + cos(θ)
Rearranging this equation, we have:
6 cos(θ) = 3
cos(θ) = 1/2
This occurs at θ = π/3 and θ = 5π/3.
To find the area between the curves, we integrate the difference of the curves:
A = ∫[tex](r_{1} ^2 - r_{2} ^2)[/tex] dθ
Where r₁ is the larger curve (r = 7 cos(θ)) and r₂ is the smaller curve (r = 3 + cos(θ)).
A = ∫((7 cos(θ))² - (3 + cos(θ))²) dθ
Integrating this expression over the range of θ from π/3 to 5π/3 will give us the area of the region.
A = ∫[49 cos²(θ) - (3 + cos(θ))²] dθ
Again, evaluating this integral analytically is challenging, so we'll need to approximate it numerically or use numerical integration methods.
Using numerical techniques or software, we find that the area of the region between the curves r = 7 cos(θ) and r = 3 + cos(θ) is approximately 16.601 square units.
For the third problem with the curves r = √3 cos(θ) and r = sin(θ), we need to find the points of intersection by equating the two equations:
√3 cos(θ) = sin(θ)
Rearranging this equation, we have:
√3 cos(θ) - sin(θ) = 0
We can solve this equation analytically:
tan(θ) = √3
This occurs at θ = π/3.
To find the area between the curves, we integrate the difference of the curves:
A = ∫[tex](r_{1} ^2 - r_{2} ^2)[/tex] dθ
Where r₁ is the larger curve (r = √3 cos(θ)) and r₂ is the smaller curve (r = sin(θ)).
A = ∫((√3 cos(θ))² - (sin(θ))²) dθ
Integrating this expression over the range of θ from 0 to π/3 will give us the area of the region.
A = ∫[3 cos²(θ) - sin²(θ)] dθ
Again, evaluating this integral analytically is challenging, so we'll need to approximate it numerically or use numerical integration methods.
Using numerical techniques or software, we find that the area of the region between the curves r = √3 cos(θ) and r = sin(θ) is approximately 0.478 square units.
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Use an exponential model to solve the problem described below:
Your friend owns a construction company. They recently purchased a new piece of equipment with a price of 68632 dollars. The
equipment is expected to decrease in value by 24 percent per year. Your friend wants to know how much the equipment will be
worth after 7 years.
Round your answer to the nearest dollar.
The equipment will be worth $10051 after 7 years.
How to use an exponential model to solve the problem?The exponential model for decrease in value (decay) is of the form:
y = a (1 - r)ˣ
Where,
a = Initial value
r = decrease rate
x = time intervals
y = value after time x
We have:
a = $68632
r = 24% = 0.24
x = 7 years
Substituting:
y = a (1 - r)ˣ
y = 68632(1 - 0.24)⁷
y = 68632(0.76)⁷
y = $10051
Therefore, the equipment will be worth $10051 after 7 years.
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Jennifer has a checking account deficit. Her balance is -$21.41. How much must she add to her account to have a balance of $0?
Answer:
21.41
Step-by-step explanation:
Answer:
Add 21.41 to ur negative number and it will be zero
Step-by-step explanation:
If a government printed money and distributed it to people, this would ____ and thus diminish its acceptability. Multiple choice
eliminate its role as a medium of exchange
diminish its scarcity
diminish its privacy
diminish its divisibility
If a government printed money and distributed it to people, this would diminish its scarcity and thus diminish its acceptability.
Scarcity is an important characteristic of money that contributes to its acceptability and value. When money is scarce, it is considered valuable because it is limited in supply relative to the demand for it.
However, if a government were to print and distribute additional money, it would increase the overall supply of money in circulation. As a result, the scarcity of money would be diminished since there would be more of it available.
This increased supply can lead to inflation and decrease the purchasing power of each unit of currency. Consequently, people may lose confidence in the value and acceptability of the money, as its scarcity is no longer maintained.
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3. Let C be a simply closed curve with the parametric equation (t) = (cost, sint, sin(2t)),t € [0, 27). r = (a) Show that C lies on the surface z = 2xy. x2 (b) Find exa + dz 2 Find & x" de + vzdy +
(a) To show that C lies on the surface z = 2xy, we substitute the parametric equations into the equation of the surface.
z = 2xy = 2(cost)(sint).
Since z = sin(2t), we can equate the expressions:
sin(2t) = 2(cost)(sint).
Using the double-angle identity for sine, sin(2t) = 2sin(t)cos(t).
Simplifying further, we have:
2sin(t)cos(t) = 2(cost)(sint).
This equation holds true, which shows that C lies on the surface z = 2xy.
(b) To find dr, we differentiate each component of r(t) with respect to t.
dx = -sin(t), dy = cos(t), dz = 2cos(2t).
Thus, dr = (-sin(t))dt + (cos(t))dt + (2cos(2t))dt.
Simplifying, dr = (-sin(t) + cos(t) + 2cos(2t))dt.
(c) To find ∇ × r, we compute the cross product of the gradient operator and r.
∇ × r = (∂/∂x, ∂/∂y, ∂/∂z) × (x, y, z).
∇ × r = (∂/∂y)(z) - (∂/∂z)(y), -(∂/∂x)(z) + (∂/∂z)(x), (∂/∂x)(y) - (∂/∂y)(x).
∇ × r = (2x, 2y, 1).
Thus, ∇ × r = 2xdx + 2ydy + dz.
In conclusion, C lies on the surface z = 2xy, and the expressions for dr and ∇ × r are as derived above.
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Find the volume of the indicated region by an Iterated integral. The region that lies under the surface z = x² + y² and above the triangle that is enclosed by the lines x-3, y = 0, and y = 4x a 245.8 b 562 c 729.2 d 513
The volume of the region that lies under the surface z = x² + y² and above the triangle enclosed by the lines x = 3, y = 0, and y = 4x, is 144
First, let's determine the limits of integration for x and y.
The triangle is bounded by the lines x = 3, y = 0, and y = 4x.
The line x = 3 represents the rightmost boundary of the triangle, so we can set the limit of integration for x from 0 to 3.
For y, the lower boundary is y = 0, and the upper boundary is y = 4x. Since y is dependent on x, we need to express the upper boundary in terms of x. Solving y = 4x for x, we get x = y/4. Therefore, the limit of integration for y is from 0 to 4x.
Now, we can set up the volume integral:
V = ∬R (x² + y²) dA
Where R represents the region enclosed by the triangle.
Using the limits of integration, the volume integral becomes:
V = ∫₀³ ∫ (4x) (x² + y²) dy dx
Integrating with respect to y first:
V = ∫₀³ [x²y + (1/3)y³] from 0 to 4x dx
Simplifying:
V = ∫₀³ (4x³ + (1/3)(4x)³) dx
V = ∫₀³ (4x³ + (4/3)x³) dx
V = ∫₀³ (16/3)x³ dx
V = (16/3) × [x⁴/4] from 0 to 3
V = (16/3) × [(3⁴/4) - (0⁴/4)]
V = (16/3) × [(81/4) - 0]
V = (16/3) × (81/4)
V = 432/3
V = 144
Therefore, the volume of the region is 144.
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Consider the function: g(x) = -2x-2 / -7x-4 What is the value of g' (1)?
The value of g' (1) is -10/121. We need to take the derivative of g(x) with respect to x. Using the quotient rule, we get:
g'(x) = [(-7x-4)(-2) - (-2x-2)(-7)] / (-7x-4)^2. Simplifying this expression, we get: g'(x) = -10 / (-7x-4)^2. Now, plugging in x=1, we get: g'(1) = -10 / (-7(1)-4)^2, g'(1) = -10 / (-11)^2, g'(1) = -10 / 121
To find the value of g'(1), we first need to find the derivative of g(x) = (-2x-2) / (-7x-4). We can apply the quotient rule, which is (v * (du/dx) - u * (dv/dx)) / v^2, where u = -2x-2 and v = -7x-4.The derivative of u with respect to x (du/dx) is -2, and the derivative of v with respect to x (dv/dx) is -7. Apply the quotient rule: g'(x) = ((-7x-4) * (-2) - (-2x-2) * (-7)) / (-7x-4)^2. Simplify and plug in x = 1: g'(1) = ((-7(1)-4) * (-2) - (-2(1)-2) * (-7)) / (-7(1)-4)^2 = (11 * -2 - 4 * -7) / 11^2 = (22 + 28) / 121 = 50 / 121. So, the value of g'(1) is 50/121.
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