a list of 5,000 players of a team needs to be saerched for the player with highest score. what is the fastest possible

The admission price for children is $2.98 and the admission price for adults is $2.05.

Let c be the admission price for children and a be the admission price for adults.

From the first day's receipts, we have the equation:

29c + 13a = 113

From the second day's receipts, we have the equation:

42c + 35a = 196

We can solve this system of equations using any method, such as substitution or elimination.

Here, we will use the substitution method.

Solving the first equation for a, we get:

a = (113 - 29c) / 13

Substituting this expression for a into the second equation, we get:

42c + 35[(113 - 29c) / 13] = 196

Multiplying both sides by 13 to eliminate the denominator, we get:

546c + 35(113 - 29c) = 2548

Expanding the parentheses, we get:

546c + 3945 - 1015c = 2548

Simplifying, we get:

-469c = -1397

Dividing both sides by -469, we get:

c = 2.98

Substituting this value for c into either of the original equations, we can solve for a.

Using the first equation:

29c + 13a = 113

29(2.98) + 13a = 113

86.42 + 13a = 113

13a = 26.58

a = 2.05

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The expectation and variance of the length of the hypotenuse are 2√2 / 3 and 2/9, respectively.

Let X be the side length of the isosceles right triangle. Then, the length of the hypotenuse is H = X√2. We want to find the expectation and variance of H.

The probability density function of X is f(x) = 2x for 0 < x < 1, and f(x) = 0 otherwise, since X is uniformly distributed on (0,1).

To find the expected value of H, we use the formula for the expected value of a function of a random variable:

E[H] = E[X√2] = √2 E[X]

To find the variance of H, we use the formula for the variance of a function of a random variable:

Var(H) = Var(X√2) = 2 Var(X)

where we have used the fact that X and √2 are constants, so their covariance is zero.

To find Var(X), we use the formula for the variance of a continuous random variable:

Var(X) = E[X^2] - (E[X])^2

We already know E[X], so we need to find E[X^2]. To do this, we integrate X^2 times the probability density function over the range (0,1):

E[X^2] = ∫[0,1] x^2 f(x) dx = ∫[0,1] 2x^3 dx = 1/2

Therefore, Var(X) = E[X^2] - (E[X])^2 = 1/2 - (2/3)^2 = 1/18.

Finally, we have:

Var(H) = 2 Var(X) = 2/9.

Therefore, the expectation and variance of the length of the hypotenuse are 2√2 / 3 and 2/9, respectively.

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Your investment of $10000 compounded continuously at 6% p.a. would be worth $14,366.00 after 6 years.

If you invest $10000 compounded continuously at 6% p.a., the formula for calculating the value of your investment after 6 years would be:

A = Pe^(rt)

Where A is the final amount, P is the principal investment amount, e is Euler's number (approximately 2.718), r is the interest rate (in decimal form), and t is the time period (in years).

Plugging in the given values, we get:

A = 10000e^(0.06*6)

A = 10000e^(0.36)

A = 10000*1.4366

A = $14,366.00

Therefore, your investment of $10000 compounded continuously at 6% p.a. would be worth $14,366.00 after 6 years.

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To find dy/dx for the given equation y^3 + x^4y^6 = 5 + ye^x, we'll differentiate both sides of the equation with respect to x using the chain rule and product rule as needed.

Differentiating y^3 + x^4y^6 = 5 + ye^x with respect to x:

Differentiating y^3 with respect to x:

(d/dx)(y^3) = 3y^2 * dy/dx

Differentiating x^4y^6 with respect to x using the product rule:

(d/dx)(x^4y^6) = 4x^3 * y^6 + x^4 * 6y^5 * dy/dx

Differentiating 5 with respect to x:

(d/dx)(5) = 0

Differentiating ye^x with respect to x using the product rule:

(d/dx)(ye^x) = e^x * dy/dx + y * e^x

Putting it all together, we have:

3y^2 * dy/dx + 4x^3 * y^6 + 6x^4 * y^5 * dy/dx = e^x * dy/dx + y * e^x

Now, let's solve for dy/dx by isolating the terms with dy/dx:

3y^2 * dy/dx + 6x^4 * y^5 * dy/dx - e^x * dy/dx = -4x^3 * y^6 - y * e^x

Factoring out dy/dx:

(3y^2 + 6x^4 * y^5 - e^x) * dy/dx = -4x^3 * y^6 - y * e^x

Dividing both sides by (3y^2 + 6x^4 * y^5 - e^x):

dy/dx = (-4x^3 * y^6 - y * e^x) / (3y^2 + 6x^4 * y^5 - e^x)

Therefore, dy/dx for the given equation y^3 + x^4y^6 = 5 + ye^x is given by the expression:

dy/dx = (-4x^3 * y^6 - y * e^x) / (3y^2 + 6x^4 * y^5 - e^x)

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The measure of angle BAD in the trapezoid (ABCD) is 100 degrees.

To find the measure of angle BAD, we can use the fact that the sum of the angles in a trapezoid is equal to 360 degrees. We know that angles B and C are opposite angles in the trapezoid, so they are congruent. Therefore, we can write:

angle B + angle C = (3x + 20) + (5x - 40) = 8x - 20

We also know that angles A and D are supplementary, since they are adjacent angles in a trapezoid. Therefore, we can write:

angle A + angle D = 180

Now we can use the fact that the sum of the angles in a trapezoid is equal to 360 to write:

angle A + angle B + angle C + angle D = 360

Substituting the expressions we have for angles B and C, and simplifying, we get:

angle A + 8x - 20 + angle A + 180 - (8x - 20) = 360

Simplifying further, we get:

2 angle A + 160 = 360

2 angle A = 200

angle A = 100

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The complete question is:

A trapezoid (ABCD) with angleB equal to (3x + 20) and angleC equal to (5x - 40)

Find the measure of angle BAD.

If the terminal point p(x, y) determined by a real number t is given then sin(t) = 13/sqrt(205), cos(t) = -6/sqrt(205), and tan(t) = -13/6.

To find sin(t), cos(t), and tan(t), we first need to determine the values of x and y. The terminal point p(x, y) is given as (−6/7, 13/7), which means that x = -6/7 and y = 13/7.

Next, we can use the Pythagorean theorem to find the length of the hypotenuse r:

r² = x² + y²

r² = (-6/7)² + (13/7)²

r² = 36/49 + 169/49

r² = 205/49

r = sqrt(205)/7

Now we can find sin(t), cos(t), and tan(t):

sin(t) = y/r = (13/7) / (sqrt(205)/7) = 13/sqrt(205)

cos(t) = x/r = (-6/7) / (sqrt(205)/7) = -6/sqrt(205)

tan(t) = y/x = (13/7) / (-6/7) = -13/6

Therefore, sin(t) = 13/sqrt(205), cos(t) = -6/sqrt(205), and tan(t) = -13/6.

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The terminal point determined by t is (-6/7, 13/7).

To find sin(t), we need to find the y-coordinate of the point on the unit circle that corresponds to t. Since the y-coordinate of the point is 13/7, and the radius of the unit circle is 1, we can use the Pythagorean theorem to find that the x-coordinate of the point is -√(1 - (13/7)²) = -√(48/49) = -4/7.

Therefore, sin(t) = y-coordinate / radius = 13/7. To find cos(t), we can use the same method to find that the x-coordinate of the point is -4/7, so cos(t) = x-coordinate / radius = -4/7. Finally, tan(t) = sin(t) / cos(t) = -(13/7)/(4/7) = -13/4.

In summary, for the terminal point determined by t (-6/7, 13/7), sin(t) = 13/7, cos(t) = -4/7, and tan(t) = -13/4. These values represent the ratios of the sides of a right triangle in standard position with hypotenuse of length 1 and one of the acute angles t.

These trigonometric functions are useful in solving various problems involving angles and distances, as well as in modeling real-world phenomena.

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The series converges for -1 < x < 1, and the interval of convergence is:

I = (-1, 1).

To find the radius of convergence, we can use the ratio test:

lim┬(n→∞)⁡|[tex]9(-1)^n n x^{2} /|9 (-1)^n nx^n[/tex]| = lim┬(n→∞)⁡|x|/|1| = |x|

The series converges if the ratio is less than 1 and diverges if it is greater than 1.

So, we need to find the values of x such that |x| < 1:

|x| < 1

Thus, the radius of convergence is R = 1.

To find the interval of convergence, we need to test the endpoints x = -1 and x = 1:

When x = -1, the series becomes:

[tex]\sum 9 (-1)^n n(-1)^n = \sum -9n[/tex]

which is divergent since it is a multiple of the harmonic series.

When x = 1, the series becomes:

[tex]\sum 9 (-1)^n n(1)^n = \sum 9n[/tex]

which is also divergent since it is a multiple of the harmonic series.

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About 68.26% of regular grade gasoline sold between $3.27 and $3.67 per gallon.

To solve this problem, we need to apply the usual normal distribution table and the formula for calculating z-score:

z = (x - μ) / σ

wherein:

First, we have to the calculate the z-ratings for the 2 given values:

z1 = (3.27 - 3.47) / 0.2 = -1

z2 = (3.67 - 3.47) / 0.2 = 1

Using the usual normal distribution table, we are able to discover the place under the curve among z1 and z2:

So, about 68.26% of regular grade gasoline sold between $3.27 and $3.67 per gallon.

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The arrangement of the number from least to greatest is: [tex]-0.35, -0.7, 1/5, 0.95[/tex].

We will start by identifying the smallest number which is -0.35. Next, we move on to the next smallest number which is -7/10.

It is important to note that negative numbers come before positive numbers when ordering from least to greatest, so, the next number in the set is 1/5, which is larger than -7/10 but smaller than 0.95

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T(t) = (i/t^(-1) + j/t^2)/√(1 + t^(-4)), N(t) = [8t^(-1)i - 2t^3j]/[8t^(-1)√(1 + t^(-4)))], and κ(t) = 6t^4(1 + t^(-4))^(-3/2). We can calculate it in the following manner.

To find T, N, and κ for the curve r(t) = t^9/9i + t^7/7j, we first find the first and second derivatives of r with respect to t:

r'(t) = t^8i + t^6j

r''(t) = 8t^7i + 6t^5j

Then we find the magnitude of r'(t):

|r'(t)| = √(t^16 + t^12) = t^8√(1 + t^(-4))

Now we can find T:

T(t) = r'(t)/|r'(t)| = (t^8i + t^6j)/[t^8√(1 + t^(-4))]

= (i/t^(-1) + j/t^2)/√(1 + t^(-4))

Next, we find N:

N(t) = T'(t)/|T'(t)| = (r''(t)/|r'(t)| - (T(t)·r''(t)/|r'(t)|)T(t))/|r''(t)/|r'(t)||

= [(8t^7i + 6t^5j)/(t^8√(1 + t^(-4))) - (t^8√(1 + t^(-4))·(8t^7i + 6t^5j)/(t^16(1 + t^(-4))))/(8t^7/√(1 + t^(-4)))|

= [8t^(-1)i - 2t^3j]/[8t^(-1)√(1 + t^(-4)))]

Finally, we find κ:

κ(t) = |N'(t)|/|r'(t)| = |(r'''(t)/|r'(t)| - (T(t)·r'''(t)/|r'(t)|)T(t) - 2(N(t)·r''(t)/|r'(t)|)N(t))/|r'(t)/|r'(t)|||

= |[(336t^5i + 180t^3j)/(t^8√(1 + t^(-4))) - (t^8√(1 + t^(-4))·(336t^5i + 180t^3j)/(t^16(1 + t^(-4))))/(8t^7/√(1 + t^(-4))) - 2[(8t^(-1)i - 2t^3j)·(8t^7i + 6t^5j)/(t^8√(1 + t^(-4)))]]/t^8√(1 + t^(-4))

= |(48t^3)/[8t^(-1)√(1 + t^(-4)))^3]|

= 6t^4(1 + t^(-4))^(-3/2)

Therefore, T(t) = (i/t^(-1) + j/t^2)/√(1 + t^(-4)), N(t) = [8t^(-1)i - 2t^3j]/[8t^(-1)√(1 + t^(-4)))], and κ(t) = 6t^4(1 + t^(-4))^(-3/2).

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Answer:24

Step-by-step explanation:

If you do 72 divided by 3 you get 24

p(3) = -23/2(3)^2 + 3/8(3) + 1/23 = 4 the polynomial satisfies the given criteria.

What is polynomial?

A polynomial is a mathematical expression that consists of variables and coefficients, which are combined using arithmetic operations such as addition, subtraction, multiplication, and non-negative integer exponents.

(a)(i) We know that a quadratic polynomial is of the form y = -ax^2 + bx + c. Using the given information, we can set up the following system of equations:

p(2) = 6:

-4a + 2b + c = 6

p(x) has a horizontal tangent at (3, 4):

p'(3) = 0 and p(3) = 4

Taking the derivative of y = -ax^2 + bx + c, we get:

y' = -2ax + b

So, p'(3) = 0 becomes:

-6a + b = 0

And p(3) = 4 becomes:

-9a + 3b + c = 4

We now have a system of three equations with three variables:

-4a + 2b + c = 6

-6a + b = 0

-9a + 3b + c = 4

(a)(ii) Setting up the augmented coefficient matrix:

| -4 2 1 | 6 |

| -6 1 0 | 0 |

| -9 3 1 | 4 |

Using Gaussian elimination, we can perform the following row operations:

R2 → R2 + (3/2)R1:

| -4 2 1 | 6 |

| 0 4 3/2 | 9 |

| -9 3 1 | 4 |

R3 → R3 - (9/4)R2:

| -4 2 1 | 6 |

| 0 4 3/2 | 9 |

| 0 -3/4 -25/4| -17/4|

R1 → R1 + R2:

| -4 6 5/2 | 15 |

| 0 4 3/2 | 9 |

| 0 -3/4 -25/4| -17/4|

R1 → (-1/4)R1:

| 1 -3/2 -5/8 | -15/4 |

| 0 4 3/2 | 9 |

| 0 -3/4 -25/4 | -17/4 |

R2 → (1/4)R2:

| 1 -3/2 -5/8 | -15/4 |

| 0 1 3/8 | 9/4 |

| 0 -3/4 -25/4 | -17/4 |

R1 → R1 + (3/2)R2:

| 1 0 1/2 | 3/4 |

| 0 1 3/8 | 9/4 |

| 0 0 -23/8 | -1/4|

R3 → (-8/23)R3:

| 1 0 1/2 | 3/4 |

| 0 1 3/8 | 9/4 |

| 0 0 1 | 1/23|

R1 → R1 - (1/2)R3:

| 1 0 0 | 5/23 |

| 0 1 3/8 | 9/4 |

| 0 0 1 | 1/23|

We can now read off the values of a, b, and c from the augmented matrix:

a = 1/(-2*1/23) = -23/2

b = 3/8

c = 1/23

Therefore, the quadratic polynomial that satisfies the given criteria is:

p(x) = -23/2x² + 3/8x + 1/23

To check that this polynomial satisfies the given criteria, we can verify that:

p(2) = -23/2(2)² + 3/8(2) + 1/23 = 6

p'(3) = -23/2(2*3) + 3/8 = 0

p(3) = -23/2(3)² + 3/8(3) + 1/23 = 4

So, the polynomial satisfies the given criteria.

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Answer: 1

Step-by-step explanation:

the two points given tell us the following information:

the difference in y coordinates, (4), will be the width of the rectangle

the difference in x coordinates, (3), will be the length of the rectangle.

haha! actually the width is the length and the length is the width lol.

anyways, the length is 1 longer than the width, since 4 -3 = 1.

For a loan amount taken by Timmy from the bank on simple interest, the interest rate advertised by the bank is equals to the 2.7% per year.

Simple interest defines to the interest calculated only based on the principal. With simple interest method, a borrower only pays interest on the principal. It is calculated by the principal amount multiplied by the interest rate, multiplied by the number of periods and then resultant is divided by 100. Formula is written as [tex]Simple \: interest = \frac{P \times r \times t}{100}[/tex]

Where, P--> principal amount

t --> time period

r -> simple interest rate

We have Timmy takes out a loan on simple interest. The amount of loan that is principal = $750

Time periods = 15 months

The received amount by him = $725

So, simple interest = 750 - 725 = $25

We have to determine the simple interest rate advertised by the bank. Using the above formula, substitute all known values in formula, 25 = [tex] \frac{ 750 × 15 × r}{12×100}[/tex]

[tex]r = \frac{ 1200× 25}{750× 15}[/tex]

= 2.66% per year

Hence, required interest rate is 2.7 % per year.

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For a normal distribution of weight of particular fruit's, the grams of weight fruit which is lighter then the heaviest 16% of fruits weight is equals to the 310.9340 g.

Z- scores used to determine percentages/probabilities/proportions related to normally distributed random variables. The z-score is a dimensionless number, and it is calculated by the formula, [tex]Z = \frac{X - \mu}{\sigma} [/tex]

where, x is the random variable

We have Mean of weight, μ = 300 grams

Standard deviations of weight,σ = 11 g

We have to determine the heaviest 16% of fruits weigh more than which grams. Now, the percentage of fruit that is heavier, p = 0.16

First, we determine the percentage of fruits that are lighter, so P = 1− 0.16 = 0.84

Now, using the distribution table the value of Z score for 84% is equals to the 0.994. So, plug all known values in above formula, [tex]0.994 = \frac{X - 300}{11}[/tex]

=> X = 11 × 0.994 + 300

=> X = 310.9340.

Hence, required weigh is 310.9340 grams.

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True. The scope of a variable refers to the segment or portion of a program where it can be accessed and utilized.

Variable scope is essential in programming because it helps maintain well-structured, organized code and prevents unintended modifications or collisions between variables with the same name in different parts of the program.

There are two primary types of variable scope: local scope and global scope. A local variable is defined within a specific function or block of code, and it can only be accessed within that particular area. Once the function or block of code is exited, the local variable ceases to exist, and its memory is freed up.

On the other hand, a global variable is accessible throughout the entire program. It is typically declared outside of any function or code block, making it available for use by any part of the code. However, using global variables can lead to potential issues, such as unintentional changes to their values and increased complexity in managing the flow of information within the program.

Understanding the scope of variables is crucial for efficient and effective programming. Proper management of variable scope promotes clean, maintainable code, and reduces the likelihood of bugs or errors resulting from variable conflicts or unintended modifications.

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Complete Question

The scope of a variable is the segment of the program in which the variable can be accessed. State whether True or False.

In programming, the increment and decrement operations can be performed using the ++ and -- operators, respectively.

However, these operators cannot be directly applied to an enumeration type, as enumeration types represent a distinct set of named constants rather than numeric values. To achieve the desired outcome, you can use a typecast to convert the enumeration type to an underlying integral type, such as int, perform the increment or decrement operation, and then convert it back to the original enumeration type.

This way, you can modify the enumeration value while preserving the context and intent of the enumeration in your code.

For example, if you have an enumeration type named "DaysOfWeek", and you want to increment the value of a variable of this type, you can use the following code:

DaysOfWeek day = DaysOfWeek.Monday;
day = (DaysOfWeek)((int)day + 1);

This code first typecasts the enumeration value to an int, adds 1 to it, and then typecasts it back to the DaysOfWeek enumeration type. Similar code can be used for decrementing the value using the -- operator.

Keep in mind that this method may not always be ideal, as it can lead to invalid enumeration values if the new value does not correspond to a named constant in the enumeration. Always ensure that you are working within the valid range of values when using this approach.

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Well based on the info given I’d say you would expect to see about 14 more.

Someone in 1955 paid only 0.2% of today's price for a loaf of bread

To find what percentage of today's price someone in 1955 paid for a loaf of bread, we need to use the concept of inflation. Inflation is the increase in the general price level of goods and services in an economy over a period of time. In other words, the cost of goods and services increases over time due to inflation.

To calculate the inflation rate, we can use the following formula:

Inflation rate = (Current price - Base price) / Base price x 100%

Here, the base price is the price of bread in 1955, and the current price is the price of bread today.

Base price = 20 cents

Current price = $2.50

Using the formula, we get:

Inflation rate = ($2.50 - $0.20) / $0.20 x 100%

Inflation rate = $2.30 / $0.20 x 100%

Inflation rate = 1150%

This means that the price of bread has increased by 1150% since 1955 due to inflation. To find out what percentage of today's price someone in 1955 paid, we can divide the 1955 price by the inflation rate and multiply by 100%.

Percentage of today's price = (Base price / Inflation rate) x 100%

Percentage of today's price = (20 cents / 1150%) x 100%

Percentage of today's price = 0.002 x 100%

Percentage of today's price = 0.2%

Therefore, someone in 1955 paid only 0.2% of today's price for a loaf of bread.

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The statement holds for n=k+1.

By mathematical induction, we have proven that 1·1!+2·2!+···+n·n!=(n+1)!−1 for all positive integers n.

What are integers?

Integers are a set of numbers that include whole numbers (positive, negative, or zero) as well as their opposites.

We will use mathematical induction to prove the statement.

Base case: Let n=1. Then the left-hand side of the equation is 1·1!=1 and the right-hand side is (1+1)!=2!-1=1. Therefore, the statement holds for n=1.

Induction hypothesis: Assume that the statement holds for some positive integer k, i.e., 1·1!+2·2!+···+k·k!=(k+1)!−1.

Inductive step: We need to show that the statement also holds for k+1, i.e., 1·1!+2·2!+···+(k+1)·(k+1)!=(k+2)!−1.

We have:

1·1!+2·2!+···+k·k!+(k+1)·(k+1)!=k!+1·1!+2·2!+···+k·k!+(k+1)·(k+1)!=k!+(k+1)!−1+(k+1)·(k+1)!=k!(k+1+1)+(k+2)!−1=(k+1)!(k+2)−1=(k+2)!−1,

where we have used the induction hypothesis in the second step and simplified in the fourth step.

Therefore, the statement holds for n=k+1.

By mathematical induction, we have proven that 1·1!+2·2!+···+n·n!=(n+1)!−1 for all positive integers n.

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Answer:

see below, answers are underlined

Step-by-step explanation:

1. Vertical angles are 2 angles that are on opposite sides of each other and are the same value.  So, 2 angles that are considered vertical angles would be MPN (5y) and LPO (95)

2. Adjacent angles are 2 angles that are right next to each other on the same line and when added, they equal 180.  So, 2 angles that are adjacent angles are MPL (5x) and MPN (5y)

3. Using what we know about adjacent and vertical angles, we can solve to find x in 2 different ways:

--> 5x=85 (vertical angles), x=17

--> 5x+95=180 (adjacent angles), x=17

4. Using what we know about adjacent and vertical angles, we can solve to find y also in 2 different ways:

-->5y=95 (vertical angles), y=19

-->5y+85=180 (adjacent angles), y=19

Hope this helps! :)

The calculated value of the number of people who had slept for less than 6 hours is 20

From the question, we have the following parameters that can be used in our computation:

The density plot

From the plot, we have

The cumulative frequency (CF) of people who had slept for less than 6 hours to be

CF = 20

This means that the number of people who had slept for less than 6 hours is 20

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The solution is, : OPTION D: NEITHER, ordered pair is a solution to the equation.

Here, we have,

The given equation is: 7x - 2y = - 5

To find a solution to this, we substitute the options and compare LHS and RHS.

OPTION A: (1, 5)

LHS = 7(1) - 2(5) = 7 - 10 = -3

RHS = - 5

LHS  RHS.

So, this option is eliminated.

OPTION B: (-1, 1)

LHS = 7(-1) - 2(1) = -7 - 2 = - 9

RHS = - 5

Again, LHS ≠ RHS.

So, this Option is eliminated as well.

OPTION C: It says both A and B. Clearly, this is eliminated as well.

This is a two variable equation. So, we need a minimum of two equations to determine the solution. Since, only one equation is given here, we use the help of options.

Therefore, the answer is: OPTION D: NEITHER, ordered pair is a solution to the equation.

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(a) The probability that the sample proportion will be within +0.03 of the population proportion is 0.7242.

(b) The probability that the sample proportion will be within +0.05 of the population proportion is 0.9312.

(a) The standard error of the sample proportion is given by:

SE = √[p(1-p)/n]

where p = population proportion, n = sample size

SE = √[0.7(1-0.7)/300] = 0.0274

To find the probability that the sample proportion will be within +0.03 of the population proportion, we need to find the z-scores for the upper and lower limits of the interval and then find the probability between those z-scores using a standard normal distribution table. The z-score for +0.03 is:

z = (0.03)/0.0274 = 1.09

The z-score for -0.03 is -1.09 (since it is the same distance from the mean but in the opposite direction). Thus, we need to find the probability between -1.09 and 1.09:

P(-1.09 < z < 1.09) = P(z < 1.09) - P(z < -1.09)

Using a standard normal distribution table, we find:

P(z < 1.09) = 0.8621

P(z < -1.09) = 0.1379

Therefore, the probability that the sample proportion will be within +0.03 of the population proportion is:

0.8621 - 0.1379 = 0.7242 (rounded to four decimal places)

(b) Using the same formula for standard error, we get:

SE = √[0.7(1-0.7)/300] = 0.0274

To find the probability that the sample proportion will be within +0.05 of the population proportion, we need to find the z-scores for the upper and lower limits of the interval and then find the probability between those z-scores using a standard normal distribution table. The z-score for +0.05 is:

z = (0.05)/0.0274 = 1.82

The z-score for -0.05 is -1.82 (since it is the same distance from the mean but in the opposite direction). Thus, we need to find the probability between -1.82 and 1.82:

P(-1.82 < z < 1.82) = P(z < 1.82) - P(z < -1.82)

Using a standard normal distribution table, we find:

P(z < 1.82) = 0.9656

P(z < -1.82) = 0.0344

Therefore, the probability that the sample proportion will be within +0.05 of the population proportion is:

0.9656 - 0.0344 = 0.9312 (rounded to four decimal places)

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Answer:

True

Step-by-step explanation:

The left side 0.3 is less than the right side 0.8, which means that the given statement is always true.

Answer:  the particular solution to the given differential equation is:

y_p(x) = u1(x)(c1 + c2x)e^(6x) + u2(x)(c1 + c2x

To find a particular solution to the given differential equation using the method of variation of parameters, we first need to find the complementary solution.

The characteristic equation of the homogeneous equation y" - 12y' + 36y = 0 is:

r^2 - 12r + 36 = 0

Factoring the equation, we have:

(r - 6)^2 = 0

This implies that the complementary solution is:

y_c(x) = (c1 + c2x)e^(6x)

Next, we find the Wronskian:

W(x) = e^(6x)

Now, we can find the particular solution using the variation of parameters. Let's assume the particular solution has the form:

y_p(x) = u1(x)(c1 + c2x)e^(6x) + u2(x)(c1 + c2x)e^(6x)

To find u1(x) and u2(x), we need to solve the following equations:

u1'(x)(c1 + c2x)e^(6x) + u2'(x)(c1 + c2x)e^(6x) = 0

u1'(x)(c1 + c2x)e^(6x) + u2'(x)(c1 + c2x)e^(6x) = 6x^2 + 49 + x^2

Differentiating the first equation with respect to x, we have:

u1''(x)(c1 + c2x)e^(6x) + u2''(x)(c1 + c2x)e^(6x) = 6 + 2x

Now, we can solve this system of equations to find u1(x) and u2(x).

From the first equation, we have:

u1'(x)(c1 + c2x)e^(6x) + u2'(x)(c1 + c2x)e^(6x) = 0

Integrating both sides with respect to x, we get:

u1(x)(c1 + c2x)e^(6x) + u2(x)(c1 + c2x)e^(6x) = A

where A is a constant of integration.

From the second equation, we have:

u1''(x)(c1 + c2x)e^(6x) + u2''(x)(c1 + c2x)e^(6x) = 6 + 2x

Simplifying, we have:

u1''(x)(c1 + c2x)e^(6x) + u2''(x)(c1 + c2x)e^(6x) = 6 + 2x

To solve this equation, we can assume that u1''(x) = 0 and u2''(x) = (6 + 2x)/(c1 + c2x)e^(6x).

Integrating u2''(x) with respect to x, we get:

u2'(x) = ∫[(6 + 2x)/(c1 + c2x)e^(6x)]dx

Integrating u2'(x) with respect to x, we get:

u2(x) = ∫[∫[(6 + 2x)/(c1 + c2x)e^(6x)]dx]dx

By evaluating these integrals, we can obtain the expressions for u1(x) and u2(x).

Finally, the particular solution to the given differential equation is:

y_p(x) = u1(x)(c1 + c2x)e^(6x) + u2(x)(c1 + c2x

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The volume of the triangular prism is 866.0 yd³

The volume of the triangular prism is given by V = Ah where

Now, we noice that in the figure, the base is an equilateral triangle with sides 10 yd.

So, its area is A = 1/2b²sinФ where

So, substituting this into the equation for the volume of the triangular prism, we have that

V = Ah

= 1/2b²sinФ × h

= 1/2b²hsinФ

Given that for the equilateral triangular base

For the pyramid

So, substituting the values of the variables into the equation, we have that

V = 1/2b²hsinФ

= 1/2(10 yd)² × 20 ydsin60°

= 1/2 × 100 yd² × 20 yd × 0.8660

= 50 yd² × 20 yd × 0.8660

= 1000 yd³ × 0.8660

= 866.0 yd³

So, the volume is 866.0 yd³

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