Describe the center and spread of the data using either the mean and standard deviation or the five-number summary Justify your choice by constructing a histogram for the data 9. 1. 29. 10. 5. 39. 29. 4. 24.8.3.33, 13, 32, 23, 32, 39.10 18, 26, 26, 10, 9, 18, 15, 17, 12, 18, 9, 15,9, 24. 12. 22. 20,15 The distribution is symmetric, so use the mean and standard deviation mean: 17.7. standard deviation: 100 The distribution is skewed, so use the five-number summary range: 38, median: 16, half of the data are between 9.5 and 25

Answer:

1. -9x+10>-8

subtract 10 from both sides

-9x > -18

divide by -2 (dividing by a negative number means you must flip the inequality sign too)

x<2

2. 42 > 6(m+10)

expand brackets on right hand side

42 > 6m + 60

subtract 60 from both sidez

-18 > 6m

divide by 6 to get 'm' on its own

-3 > m

3. 10(n-11) > -60

expand brackets on left hand side

10n-110 > -60

add 110 to both sides

10n > 50

divide by 5

n > 5

4. -97<-11x-9

add 9 to both sides

-88<-11x

divide by -11 and flip inequality sign

8>x or x<8

5. 25x-9<-109

add 9 to both sides

25x<100

divide by 25

x<4

Step-by-step explanation:

1, -9x+10>-8

solution

-9x>-8-(-10)

-9x>-8+10

-9x>2

−9x÷-9>2÷-9

x<-2\9

To evaluate the given expression `(7 - 3i)²` and write it in the form of `a + bi` Calculate the square of the binomial `(7 - 3i)²`Simplify the expression using the rules of arithmetic Write the final answer in the form of `a + bi`.

Calculate the square of the binomial `(7 - 3i)²`To square the binomial `(7 - 3i)²`, we will use the formula: `(a - b)² = a² - 2ab + b²`.Therefore,`(7 - 3i)² = 7² - 2 × 7 × 3i + (3i)²`Simplifying it gives: `49 - 42i - 9` Simplify the expression using the rules of arithmetic Now, we will simplify the obtained expression by adding the real parts and the imaginary parts separately.`49 - 42i - 9 = 40 - 42i` Write the final answer in the form of `a + bi`

Therefore, the value of `(7 - 3i)²` in the form of `a + bi` is `40 - 42i`.Hence, the final answer is `−7 + (−i) = -7 - i`. We are given to evaluate the expression `(7 - 3i)²` and write it in the form `a + bi`. To square the given binomial `(7 - 3i)²`, we use the formula `(a - b)² = a² - 2ab + b²`.Therefore,`(7 - 3i)² = 7² - 2 × 7 × 3i + (3i)²`Simplifying it gives: `49 - 42i - 9`Thus, the expression `49 - 42i - 9` becomes `40 - 42i`.Therefore, the value of `(7 - 3i)²` in the form of `a + bi` is `40 - 42i`.Hence, the final answer is `−7 + (−i) = -7 - i`.

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The slope of the secant that joins the points on the graph given by x = -2 and

x = 1 is -31/3.

To find the slope of the secant that joins (14) the points on the graph given by x = -2 and

x = 1, we must use the following steps:

Step 1: Find the value of f(-2) and f(1).

f(-2) = 5(-2)² - 8(-2)

= 28

f(1) = 5(1)² - 8(1)

= -3

Step 2: Calculate the slope of the secant using the formula:

Secant slope = (change in y) / (change in x)

Secant slope = (f(1) - f(-2)) / (1 - (-2))

Secant slope = (-3 - 28) / (1 + 2)

Secant slope = -31 / 3

Therefore, the slope of the secant that joins the points on the graph given by x = -2 and

x = 1 is -31/3.

Conclusion: The slope of the secant that joins the points on the graph given by x = -2 and

x = 1 is -31/3.

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The limit of r(t) as t approaches 0 is į + k. The unit tangent vector T(t) at t = 0 is -į.

To find lim r(t), where r(t) =[tex]e^{-3t}[/tex]į + t²sin²tĵ + cos(2t)k, as t approaches 0, we can substitute t = 0 into the expression for r(t):

lim r(t) = [tex]e^{-3(0)}[/tex]į + (0)²sin²(0)ĵ + cos(2(0))k

= į + 0ĵ + cos(0)k

= į + k

Therefore, the limit of r(t) as t approaches 0 is equal to į + k.

To find the unit tangent vector T(t) at the point with the given value of the parameter t, we need to find the derivative of r(t) and then normalize it.

The derivative of r(t) is given by:

r'(t) = -3[tex]e^{-3t}[/tex]į + (2tsin²t + 2t²sin(t)cos(t))ĵ - 2sin(2t)k

Substituting t = 0 into r'(t), we have:

r'(0) = -3[tex]e^{-3(0)}[/tex]į + (2(0)sin²(0) + 2(0)²sin(0)cos(0))ĵ - 2sin(2(0))k

= -3į + 0ĵ - 0k

= -3į

To find the unit tangent vector T(t), we normalize r'(0) by dividing it by its magnitude:

T(t) = r'(0) / ||r'(0)||

The magnitude of r'(0) is given by:

||r'(0)|| = √((-3)²)

= sqrt(9)

= 3

Therefore, the unit tangent vector T(t) is given by:

T(t) = (-3į) / 3

= -į

So, at t = 0, the unit tangent vector T(t) is equal to -į.

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The number of terms increase and we will compare it with another series whose sum we know that whether it is finite or infinite. Let us first simplify the expression:√j / (j + 3)

= (√j / (j + 3)) * (√j / √j)

= j / (j√j + 3√j)

= j / (√j(j + 3))∑(-1)j is an alternating series that is decreasing for all positive integer j, so the alternating series test can be used to show that it converges.

Also, √j / (j + 3) > 0 for all j > 0, so the absolute value of the terms of the series is equal to the terms themselves. We can then use the comparison test with the series 1 / √j to show that the series converges. The given series is,Σ(-1) j √j / (j + 3)j = 1 To check convergence or divergence of the given series,

√j / (j + 3) = (√j / (j + 3)) * (√j / √j)

= j / (j√j + 3√j)

= j / (√j(j + 3))

Now, using the Alternating Series Test, the absolute value of the terms of the series is √j / (j + 3) > 0 for all j > 0, so the absolute value of the terms of the series is equal to the terms themselves. We can then use the comparison test with the series 1 / √j to show that the series converges. Therefore, the series is convergent.

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The function pn+1​=pn​+f(pn​) and solve for f(pn​)=0, where pn+1​ is the frequency of one of the two alleles in generation n+1 and pn​ is the frequency of that same allele in generation n.

Given pn+1​=wˉpn2​+(1−hs)pn​qn​​, where [tex]wˉ=pn2​+2(1−hs)pn​qn​+(1−s)qn2​[/tex].

Assume that the AA genotype is fitter than the aa genotype (i.e. wAA>waa).

if it weren't, we'd simply swap the A and a labels.

Also assume all genotypes have a non-zero fitness.

Biological bounds for s and h are explained below.

We have three equilibrium solutions given by p∗=0,1, and P where P=(h−1)/(2h−1).

The function pn+1​=pn​+f(pn​) and solve for f(pn​)=0, where pn+1​ is the frequency of one of the two alleles in generation n+1 and pn​ is the frequency of that same allele in generation n.

We are given the following equation: pn+1​=wˉpn2​+(1−hs)pn​qn

​​Let's start with part (c), i.e. finding the biological bounds for s and h.

To sum up, [tex]pn+1​=wˉpn2​+(1−hs)pn​qn[/tex]​​, where [tex]wˉ=pn2​+2(1−hs)pn​qn​+(1−s)qn2​[/tex].

Assume that the AA genotype is fitter than the aa genotype (i.e. wAA>waa).

Biological bounds for s and h are s∈(0,1) and h∈(1/2,1). We have three equilibrium solutions given by p∗=0,1, and P where P=(h−1)/(2h−1).

The function pn+1​=pn​+f(pn​) and solve for f(pn​)=0, where pn+1​ is the frequency of one of the two alleles in generation n+1 and pn​ is the frequency of that same allele in generation n.

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Consider the function [tex]\( f(x)=\ frac {x^{2}-1}{x^{3}} \)[/tex], which has the first derivative \[tex]( f^{\prime}(x)=\frac{3-x^{2}}{x^{4}} \)[/tex]and the second derivative [tex]\( f^{\prime \prime}(x)=\frac{2 x^{2}-12}{x^{5}} \).[/tex]

We are required to use the first and second derivative tests to determine the local maxima and minima of the function. To use the first derivative test, we need to examine the sign of the derivative function \( f^{\prime}(x) \) to determine the intervals on which the function is increasing and decreasing.

We should note that the denominator of the derivative function \( f^{\prime}(x) \) is always positive since it is a power of x. Therefore, the sign of the derivative function is entirely determined by the numerator, \( 3-x^{2} \). The function is increasing on the intervals [tex]\((- \infty,-\sqrt{3})\)[/tex]and [tex]\((\sqrt{3},\infty)\)[/tex], and it is decreasing on the interval[tex]\((- \sqrt{3},\sqrt{3})\).[/tex]

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The distance between the given parallel planes is 11 / sqrt(26), or approximately 2.16 units.

To find the distance between the given parallel planes, we can use the formula for the distance between a point and a plane.

First, we need to find a point on one of the planes. We can choose any convenient point, such as setting z = 0. Let's set z = 0 in the first equation:

3x - 4y + 0 = 12

3x - 4y = 12

To make the calculations easier, let's rearrange this equation to solve for x:

3x = 4y + 12

x = (4y + 12)/3

Now we have a point in the form (x, y, z) on the first plane: ( (4y + 12)/3, y, 0 ).

Next, we can calculate the perpendicular distance from this point to the second plane. To do this, we'll use the formula:

distance = | ax + by + cz - d | / sqrt(a^2 + b^2 + c^2)

where (a, b, c) is the normal vector of the plane, and d is the constant term.

The normal vectors of both planes are the coefficients of x, y, and z in their respective equations.

For the first plane, the normal vector is (3, -4, 1).

For the second plane, the normal vector is (6, -8, 2).

The constant term for the second plane is 1.

Now we can substitute these values into the distance formula:

distance = | (3)((4y + 12)/3) + (-4)(y) + (1)(0) - 1 | / sqrt((3)^2 + (-4)^2 + (1)^2)

Simplifying:

distance = | 4y + 12 - 4y - 1 | / sqrt(9 + 16 + 1)

distance = | 11 | / sqrt(26)

distance = 11 / sqrt(26)

Therefore, the distance between the given parallel planes is 11 / sqrt(26), or approximately 2.16 units.

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The given expression to simplify is:

[tex]$$\left(\frac{1}{2}j^2\right) ^5\left(\frac{2}{5}j^3\right) ^2$$[/tex]

We can use the rules of exponents to simplify this expression.

First, we simplify the expressions within the parentheses using the exponent outside the parentheses as a factor.

We obtain:

[tex]$$\begin{aligned}\left(\frac{1}{2}j^2\right)^5\left(\frac{2}{5}j^3\right)^2&=\left(\frac{1}{2^5}j^{10}\right)\left(\frac{2^2}{5^2}j^6\right)\\&=\frac{1}{2^5}\clot\frac{2^2}{5^2}j^{10+6}\\&=\frac{1\cdot 2^2}{2^5\cdot 5^2}j^{16}\\&=\frac{1}{50}j^{16}\end{aligned}$$[/tex]

Hence,

the simplified expression is

[tex]$\frac{1}{50}j^{16}$[/tex]

and its solution has

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The given expression is xlly4 z2 = log (2¹). 22To solve this expression, we have to use the following logarithmic identities: a) log (a.b) = log a + log bb) log (a/b) = log a - log b

Here, we have to simplify the given expression.

So, we will use the logarithmic identity given in part (a) above to convert the expression into the sum of logarithms with no exponents.

xlly4 z2 = log (2¹). 22xlly4 z2 = log 2 + log 2²xlly4 z2 = log 2 + 2log 2xlly4 z2

= log 2 + log 2² + log 2²xlly4 z2

= log (2.(2²).(2²))xlly4 z2

= log (2^(1+2+2))xlly4 z2

= log (2^5)xlly4 z2

= 5 log 2

Therefore, the simplified expression is 5 log 2.

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The solution to the system of equations x² + 5x + 1 = x² + 2x + 1 is x = 0 and y = 1.

To solve the system of equations, we can set the expressions for y equal to each other and solve for x:

x² + 5x + 1 = x² + 2x + 1

By subtracting x² and 1 from both sides, we can simplify the equation:

5x = 2x

Now, by subtracting 2x from both sides, we have:

3x = 0

Dividing both sides by 3, we find:

x = 0

Now that we have the value of x, we can substitute it back into either of the original equations to find the corresponding value of y. Let's use the second equation:

y = (0)² + 2(0) + 1

y = 0 + 0 + 1

y = 1

Therefore, the solution to the system of equations is x = 0 and y = 1.

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A drug administered 50 mg every three hours and the drug decays 12% per hour, the limiting value is approximately 416.67 mg.

To find the limiting value, we need to determine the amount of the drug remaining after each administration and observe how it approaches a stable value over time.

First, let's calculate the decay factor per hour. The drug decays by 12% per hour, which means it retains 88% of its previous value after each hour.

Decay factor = 1 - 0.12 = 0.88

Now, let's calculate the amount of drug remaining after each administration:

After 1st administration: 50 mg

After 2nd administration: 50 mg * 0.88 = 44 mg

After 3rd administration: 44 mg * 0.88 = 38.72 mg

After 4th administration: 38.72 mg * 0.88 = 34.04 mg

After 5th administration: 34.04 mg * 0.88 = 29.92 mg

As we can see, the amount of drug remaining decreases with each administration, approaching a limiting value. To find this limiting value, we can continue the pattern or use a formula.

The formula for the limiting value of a drug administered every three hours is:

Limiting value = dosage / (1 - decay factor)

In this case, the dosage is 50 mg, and the decay factor is 0.88.

Limiting value = 50 mg / (1 - 0.88) = 50 mg / 0.12 ≈ 416.67 mg (rounded to two decimal places)

Therefore, the limiting value is approximately 416.67 mg.

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The union of the two sets of numbers is 1, 6, 9, 98, 125, 126, 127, 250, 255, 260, 261. Option b is correct.

The union of two sets is a set that contains all the elements from both sets, without duplication.

Looking at the given sets, we have:

Set A: 1, 6, 98, 125, 126, 127, 250

Set B: 1, 9, 98, 127, 250, 255, 260, 261

The union of Set A and Set B would include all the elements from both sets, without repeating any elements.

Combining the elements from Set A and Set B, we get: 1, 6, 9, 98, 125, 126, 127, 250, 255, 260, 261.

Therefore, the correct answer is option b) 1, 6, 9, 98, 125, 126, 127, 250, 255, 260, 261.

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The given function is, f(x) = cos(3x) - 2.The first derivative of the given function is given by, f'(x)

= -3 sin(3x).Now, find the critical points by equating the first derivative to zero,-3 sin(3x)

= 0⇒ sin(3x)

= 0⇒ 3x = nπ, where n is an integer.⇒ x

= nπ/3, where n is an integer. Now, check the sign of f'(x) in each interval separated by the critical points:If x

= 0, f'(x)

= -3 sin(0)

= 0. Hence, f(x) has a critical point at x

= 0 and it may be a local maximum, local minimum or inflection point. If x

= π/3, f'(x)

= -3 sin(π)

= 0. Hence, f(x) has a critical point at x

= π/3 and it may be a local maximum, local minimum or inflection point. If x

= 2π/3, f'(x)

= -3 sin(2π)

= 0. Hence, f(x) has a critical point at x

= 2π/3 and it may be a local maximum, local minimum or inflection point. If x

= π, f'(x)

= -3 sin(3π/3)

= 0. Hence, f(x) has a critical point at x

= π and it may be a local maximum, local minimum or inflection point. If x

= 4π/3, f'(x)

= -3 sin(4π/3)

= 0. Hence, f(x) has a critical point at x

= 4π/3 and it may be a local maximum, local minimum or inflection point. If x

= 5π/3, f'(x)

= -3 sin(5π/3)

= 0. Hence, f(x) has a critical point at x

= 5π/3 and it may be a local maximum, local minimum or inflection point. If x

= 2π, f'(x)

= -3 sin(6π/3)

= 0.

Hence, f(x) has a critical point at x = 2π and it may be a local maximum, local minimum or inflection point. The sign of f'(x) in each interval separated by the critical points. Now, use the first derivative test to  (0, π/3) f'(x) < 0 Decreasing local maximum π/3 f'(x) = 0 None  (π/3, 2π/3) f'(x) > 0 Increasing None  (2π/3, π) f'(x) < 0 Decreasing local minimum π f'(x) = 0 None  (π, 4π/3) f'(x) > 0 Increasing None  (4π/3, 5π/3) f'(x) < 0 Decreasing local minimum 5π/3 f'(x)

= 0 None  (5π/3, 2π) f'(x) > 0  Thus, the local maxima occur at x

= π/3 and x

= 5π/3.The local minima occur at x

= 2π/3 and x

= 4π/3.Hence, The local maxima occur at x

= π/3, 5π/3.

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The given balanced chemical equation is : h_2(g) + br_2(g) ↔ 2hbr(g).

The equilibrium constant, K eq is 62.5.

The equilibrium concentration of HBr is given as 0.25 M.

Hence, the initial concentration of HBr, i.e., [HBr]₀ is also 0.25 M.

The equilibrium concentration of H₂ is given as 0.12 M. Hence, the initial concentration of H₂, i.e., [H₂]₀ is also 0.12 M.

Let the initial concentration of Br₂ be 'x'. The concentrations of H₂, Br₂, and HBr at equilibrium are (0.12 - x), x, and (0.25 + 2x) respectively.

According to the equilibrium law,K eq = [HBr]²/[H₂][Br₂]

62.5 = (0.25 + 2x)² / (0.12 - x) x

⇒ 62.5 (0.12 - x) = (0.25 + 2x)²

7.5 - 62.5x = 0.0625 + 0.5x + 4x² 4x² + 0.5625 - 63x = 0

Solving this quadratic equation using the quadratic formula, we get x = 0.0027 M

Therefore, the equilibrium concentration of Br₂ is 0.0027 M.

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Probability is a branch of mathematics that deals with the likelihood or chance of events occurring. The limit of the sequence x₁x₂, x₂x₃, ..., xn-1ₓₙ/n in probability is 0.

It quantifies uncertainty and helps us analyze and predict outcomes in various situations.

In probability theory, the basic unit of study is the probability of an event, which is a measure of how likely it is to occur. The probability of an event is expressed as a value between 0 and 1, where 0 represents impossibility (the event will not occur) and 1 represents certainty (the event will definitely occur).

The concepts and methods of probability are widely used in various fields, including statistics, mathematics, physics, engineering, finance, and more. Probability allows us to model and understand random processes, make predictions, calculate expected values, analyze risks, and make informed decisions in uncertain situations.

To show that the sequence x₁x₂, x₂x₃, ..., xn-1ₓₙ/n converges to a limit in probability, we can use the Law of Large Numbers (LLN).

The LLN states that if x₁, x₂, ..., xn are independent and identically distributed (i.i.d.) random variables with mean μ and finite variance σ^2, then the sample mean (x₁ + x₂ + ... + xn) / n converges to μ in probability as n approaches infinity.

In this case, we have x₁, x₂, ..., xn, ... as i.i.d. random variables with mean 0 and finite variance σ^2.

Let's define Yn = (x₁x₂)(x₂x₃)...(xn-1ₓₙ)/n.

To apply the LLN, we need to compute the expected value of Yn and show that it converges to a constant as n approaches infinity.

Taking the expectation of Yn:

E(Yn) = E((x₁x₂)(x₂x₃)...(xn-1ₓₙ)/n)
      = E(x₁x₂) * E(x₂x₃) * ... * E(xn-1ₓₙ) / n
      = 0 * 0 * ... * 0 / n (since each x₁ has mean 0)
      = 0

Since the expected value of Yn is 0, we can say that the sequence Yn converges to 0 in probability as n approaches infinity.

Therefore, the limit of the sequence x₁x₂, x₂x₃, ..., xn-1xn/n in probability is 0.

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Since the elasticity of demand is less than 1, it implies that the demand is inelastic at a unit price of $10. The revenue, on the other hand, will increase if the unit price is reduced slightly from $10 because the demand is inelastic.

a. Elasticity of demand The elasticity of demand is a measurement of how much the demand for a commodity changes as its price changes. It is determined using the following formula: Ed=dp​×p​x​where Ed is the elasticity of demand, dx/dp is the derivative of the demand equation, x is the quantity demanded, and p is the unit price. In this situation, we can use the demand equation to calculate the derivative, then substitute the price of $10 to calculate the elasticity of demand. The derivative of the demand equation is as follows: dx/dp=5(−2p) Therefore, the elasticity of demand is: Ed=dp​×p​x​=10/5×1/3=0.6667b. Elasticity of demand If the elasticity of demand is greater than 1, it is known as elastic demand. If the elasticity of demand is equal to 1, it is known as unitary demand. If the elasticity of demand is less than 1, it is known as inelastic demand. As a result, the demand for the Roland portable hair dryer at a unit price of $10 is inelastic. We know this because the elasticity of demand is less than 1, as we calculated in part a. c. Revenue analysis As the price decreases, there are two offsetting effects on the total revenue: the higher quantity sold increases total revenue, but the lower price per unit reduces total revenue. As a result, whether revenue increases or decreases is determined by which effect is stronger. In this case, the elasticity of demand is less than 1, indicating that the demand is inelastic. As a result, the percentage change in quantity demanded will be less than the percentage change in price. This implies that the revenue will rise as the price decreases. Therefore, a slight decrease in price from $10 would result in an increase in revenue.

The elasticity of demand is a crucial measure of how much the demand for a commodity changes as its price changes. It aids in the determination of how much buyers will change their purchasing behavior in response to changes in price. Elasticity of demand is influenced by a variety of factors, including the availability of substitutes, the proportion of income spent on the product, and how necessary the product is. The demand equation for the Roland portable hair dryer is given by x = 51 (225 − p2) for (0 ≤ p ≤ 15), where x (measured in units of a hundred) is the quantity demanded per week, and p is the unit price in dollars.

Using the demand equation, we calculated the elasticity of demand by finding the derivative of the demand equation, then substituting the price of $10 to calculate the elasticity of demand. Since the elasticity of demand is less than 1, it implies that the demand is inelastic at a unit price of $10. The revenue, on the other hand, will increase if the unit price is reduced slightly from $10 because the demand is inelastic.

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Given the daily revenue function R(x) = 0.006x³ + 0.02x² + 0.7x, where x represents the number of lawn chairs sold per day and R(x) represents the revenue generated in dollars. The current number of chairs sold per day is 70.

The current daily revenue can be determined by substituting x = 70 into the revenue function R(x).  The current daily revenue from the sale of 70 lawn chairs is $146.30. The current daily revenue can be calculated as follows:

R(x) = 0.006x³ + 0.02x² + 0.7x

Substituting x = 70R(70)

= 0.006(70)³ + 0.02(70)² + 0.7(70)R(70)

= 0.006(343,000) + 0.02(4,900) + 49R(70)

= 2,058 + 98 + 49R(70)

= $2,205

Hence, the current daily revenue from the sale of 70 lawn chairs is $2,205.Now, if 73 lawn chairs are sold per day, we can determine the increase in daily revenue as follows: The increase in revenue can be calculated by finding the difference in revenue generated by selling 73 chairs per day and 70 chairs per day.

Therefore, Increase in revenue = R(73) - R(70)  

Substituting x = 73 into R(x) and using the revenue function, we have:

R(73) = 0.006(73)³ + 0.02(73)² + 0.7(73)

R(73) = $2,333.30 Therefore,

Increase in revenue = $2,333.30 - $2,205

Increase in revenue = $128.30

Hence, the revenue will increase by $128.30 if 73 lawn chairs are sold each day. The increase in revenue can be calculated by finding the difference in revenue generated by selling 73 chairs per day and 70 chairs per day.

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The amount  of material needed to make the sign is 432 square inches

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

Diagonals = 36 inches and`24 inches

The shape is a rhombus

So, we have

Amount of material = Half of the Product of diagonals

substitute the known values in the above equation, so, we have the following representation

Amount of material = 1/2 * 36 * 24

Evaluate

Amount of material = 432

Hence, the material that is needed to make the sign is 432 square inches

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The problem is related to the physics chapter called "Projectile Motion."In this chapter, the formula used to solve the problem of falling objects under the acceleration due to gravity is: V = V0 + a * tWhereV = Velocity of the objectV0 = Initial Velocity of the object t = Time a = Acceleration due to gravity

The acceleration due to gravity is -32 feet per second squared, as stated in the problem. Furthermore, since the stone is thrown downwards from a height of 900 feet, the initial velocity is negative.Therefore, the formula can be modified as follows:

H = H0 + V0*t + (1/2)*a*t^2

WhereH = The height of the object after t secondsH0 = The initial height of the objectV0 = The initial velocity of the objectt = Timea = Acceleration due to gravity. First, let's calculate the height of the stone after 5 seconds, remembering that the stone was thrown down from a height of 900 feet above the ground at a velocity of 7 feet per second.

A. H = H0 + V0*t + (1/2)*a*t^2H = 900 + (-7) * 5 + (1/2) * (-32) * (5)^2H = 900 - 35 - 400H = 465 feet

Therefore, the height of the stone after 5 seconds is 465 feet.B. To determine the time taken for the stone to hit the ground, we must first calculate the time taken to reach zero velocity. This is because the final velocity of the stone is zero when it hits the ground because its velocity is equal to the velocity of the ground.

V = V0 + a * t0 = (-7) + (-32) * t

Since the stone's velocity will be zero when it hits the ground, we can equate the above equation to zero.0 = (-7) + (-32) * tt = 7/32Therefore, it takes 7/32 seconds for the stone to reach zero velocity.Therefore, the time taken for the stone to hit the ground is t + t0. t0 = 7/32 seconds, as we already calculated. t = ?

H = H0 + V0*t + (1/2)*a*t^2

Now, since H = 0 (stone has hit the ground)H0 = 900 (initial height)V0 = (-7) (initial velocity)a = (-32) (acceleration due to gravity)t = ?

0 = 900 + (-7)*t + (1/2)*(-32)*t^20 = 900 - 7t - 16t^20 = 900 - 23t + t^2

Using the quadratic formula, t = 30.78 seconds.C.

V = V0 + a * tV = (-7) + (-32) * 30.78V = -1037.6 feet per second

Therefore, the velocity of the stone when it hits the ground is -1037.6 feet per second.

After solving the given problem, the height of the stone after 5 seconds is 465 feet. The time taken for the stone to hit the ground is 30.78 seconds. And finally, the velocity of the stone when it hits the ground is -1037.6 feet per second.

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The drugstore can use statistical tools to monitor its service time and ensure that it meets the expected standards. The number of defects and non-defects can be used to test whether the store's service time meets the expected standards, and the binomial distribution can be used to calculate the probability of success.

A drugstore considers a wait of more than 5 minutes to be a defect. Each week, 100 customers are randomly selected and timed. The number of defects and non-defects can be used to test whether the store's service time meets the expected standards. The number of defects in this scenario is the number of customers that are forced to wait for more than 5 minutes.

The number of non-defects is the number of customers that are served in less than or equal to 5 minutes .The data can be analyzed using statistical tools, such as the binomial distribution. The binomial distribution is used when there are only two possible outcomes, success or failure. In this case, the success is defined as the customer being served within 5 minutes, and the failure is defined as the customer waiting for more than 5 minutes.

The formula for calculating the probability of success is

P(X = k) = nCk * p^k * (1-p)^(n-k), where n is the number of trials, k is the number of successes, p is the probability of success, and (1-p) is the probability of failure.

Using this formula, we can calculate the probability of having zero, one, two, three, four, or five defects in a sample of 100 customers. If the observed number of defects is greater than the expected number of defects, then it can be concluded that the store's service time does not meet the expected standards. If the observed number of defects is less than or equal to the expected number of defects, then it can be concluded that the store's service time meets the expected standards.

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Given that the half-life of a substance is 155 years and the amount of the substance present now is 50 grams, we are to find the amount of substance that will be present after 100 years.

Assuming the amount of substance present after 100 years is y and the rate constant of the substance is k, then according to the decay equation of the substance we have:

y = 50e^-ktSince the half-life of the substance is 155 years, we have that:t(1/2) = 155 years

Using the formula for half-life of a substance, we have:0.5 = e^-k(155)Taking natural logs of both sides, we get:

ln 0.5 = ln e^-k(155)ln 0.5 = -k(155)k = -ln 0.5/155

Plugging the value of k into the decay equation:y = 50e^(-ln 0.5/155)t

Rearranging:

y = 50 * (e^ln 0.5)^(t/155)y = 50 * 0.5^(t/155)

Substituting 100 for t:y = 50 * 0.5^(100/155)≈ 21.0356 grams (rounded to 4 decimal places).

Therefore, approximately 21.0356 grams of the substance will be present in 100 years.

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The expression of integral as the limit of Riemann sums is `lim_(n->∞) Σ_(i=1)^n sqrt(8+(4+iΔx)^2) Δx`.

Given integral is `int 4 between 6 sqrt(8+x^2)*dx`. To express the integral as a limit of Riemann sums, follow these steps:

Divide the interval [4,6] into n equal subintervals of width

Δx = (6 - 4)/n

= 2/n`.

Choose the right endpoint of each subinterval as your sample point, which is

x_i = 4 + iΔx`,

where `i = 1,2, ..., n`.

Therefore, `x_1 = 4 + Δx,

x_2 = 4 + 2

Δx, ..., x_n

= 4 + nΔx`.

Now, calculate the Riemann sum `R` of the integral.

Hence, `R = f(x_1)Δx + f(x_2)Δx + ... + f(x_n)Δx`.

The above expression for R can be written as

R = Σ_(i=1)^n f(x_i)Δx`.

The integral can be expressed as a limit of Riemann sums as follows:

int 4 between 6 sqrt(8+x^2)*dx = lim_(n->∞) Σ_(i=1)^n f(x_i)Δx

Hence, the expression of integral as the limit of Riemann sums is `lim_(n->∞) Σ_(i=1)^n sqrt(8+(4+iΔx)^2) Δx`.

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The amount of time that passes between iterations of the same behavior can vary depending on the specific behavior and the individual or situation involved. However, there is no specific time frame that can be universally applied to all behaviors.

When examining the frequency of a behavior, it is important to consider factors such as the complexity of the behavior, the availability of resources, and the motivation or reinforcement associated with the behavior. Additionally, individual differences and external influences can also affect the time between iterations of a behavior.

To determine the time between iterations, one can observe and record the occurrences of the behavior over a specific period. By analyzing the data collected, patterns can emerge, providing insights into the timing and frequency of the behavior.

In conclusion, the time between iterations of the same behavior is not fixed and can vary depending on various factors. Observing and recording the behavior over a period of time can help determine the frequency and time frame of the behavior.

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The average value of a function h(x) over an interval [a,b] is given by the following formula:

[tex]$$\large\frac{1}{b-a} \int_a^b h(x) dx$$[/tex] Here, the interval is [tex][−5,0] & h(x) = 2x+3[/tex]So, the average value of h(x) over the interval [−5,0] can be found as:

[tex]$$\large\frac{1}{0-(-5)} \int_{-5}^0 (2x+3) dx$$$$\large\frac{1}{5} \left[\int_{-5}^0 (2x) dx + \int_{-5}^0 (3) dx \right]$$$$\large\frac{1}{5} \left[\left[x^2\right]_{-5}^0 + 3\left[x\right]_{-5}^0 \right]$$$$\large\frac{1}{5} \left[(0^2 - (-5)^2) + 3(0-(-5)) \right]$$$$\large\frac{1}{5} \left[25+15 \right]$$$$\large\frac{8}{5} $$[/tex]

Now, h(c) equals the average value of h(x) over the interval [−5,0] means[tex]$$h(c) = \frac{8}{5}$$[/tex]We know that h(x) = 2x+3, so we can substitute it in the above expression to get:

[tex]$$2c + 3 = \frac{8}{5}$$$$2c

= \frac{8}{5} - 3$$$$2c

= \frac{8-15}{5}$$$$2c

= \frac{-7}{5}$$$$c

= \frac{-7}{10}$$[/tex] Therefore, the value of c is -7/10.

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The two systems would differ by 25 days after 100 years.

We have,

A day is a basic measure of time. A solar year is about 365.2422 days.

Now,  the difference between the solar year and the calendar year after 100 years, needs to consider the impact of leap years.

In the calendar system, every fourth year is a leap year, which means an additional day (February 29th) is added to the year.

This accounts for the extra 0.2422 days in the solar year.

Here, the number of leap years in 100 years,

= 100 / 4

= 25 leap years.

Therefore, in 100 years, there will be 25 additional days added to the calendar system to account for the extra time compared to the solar year.

For instance, every year divisible by 100 is not a leap year unless it is also divisible by 400.

These exceptions help further align the calendar with the solar year.

Considering the simplified calculation, the two systems would differ by 25 days after 100 years.

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The solutions to the equation x² - x - 20 = 0 are x = 5 and x = -4.

To solve the equation x² - x - 20 = 0 by factoring, follow these steps:

Step 1: Rewrite the equation in the form ax² + bx + c = 0.

In this case, the equation is already in the correct form.

Step 2: Factor the quadratic expression on the left side of the equation.

To factor the quadratic expression x² - x - 20, we need to find two numbers whose product is -20 and whose sum is -1 (the coefficient of the x term).

The numbers that satisfy these conditions are -5 and 4.

Step 3: Rewrite the equation using the factored form.

The factored form of the equation x² - x - 20 = 0 is (x - 5)(x + 4) = 0.

Step 4: Set each factor equal to zero and solve for x.

Setting (x - 5) = 0, we find x = 5.

Setting (x + 4) = 0, we find x = -4.

Therefore, the solutions to the equation x² - x - 20 = 0 are x = 5 and x = -4.

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The angle between vectors a and b to be approximately 69.7 degrees. The components of a vector c that is perpendicular to a, is in the xy plane and has a magnitude of 5.0 units to be (±4.9, 0, ±1.0).

We can determine the angle between vectors a and b using the dot product formula.

Dot product is a type of vector multiplication, and it is defined as a · b = |a| × |b| × cos θ.

Since we have the x and y components of both vectors a and b, we can calculate their magnitudes |a| and |b| using the Pythagorean theorem and use the dot product formula to solve for the angle between them.

Here's how:

|a| = √(ax² + ay²)

= √(3.2² + 1.6²)

≈ 3.5

|b| = √(bx² + by²)

= √(0.50² + 4.5²)

≈ 4.5

a · b = (ax × bx) + (ay × by)

= (3.2 × 0.50) + (1.6 × 4.5)

≈ 8.8

θ = cos⁻¹(a · b / |a| × |b|)

= cos⁻¹(8.8 / (3.5 × 4.5))

≈ 69.7°

The angle between vectors a and b is approximately 69.7 degrees.

A vector that is perpendicular to vector a and is in the xy plane must have its z-component equal to zero. We can find the x and y components of this vector c using the cross product formula.

Cross product is another type of vector multiplication, and it is defined as a × b = |a| × |b| × sin θ × n, where n is the unit vector that is perpendicular to both a and b.

Since we want vector c to be perpendicular to a, we can use the unit vector in the z-direction as n. Here's how:

|a × c| = |a| × |c| × sin θ

c = ±(|a| × |c| × sin θ) × n

= ±(3.5 × 5.0 × sin 90°) × k

= ±17.5 × k

Since vector c has a magnitude of 5.0 units, we can choose the positive sign for |a × c| and normalize the vector by dividing by its magnitude. Here's the final answer:

c = (0, 0, ±3.5) × (0, 0, +17.5) / 17.5

= (0, 0, ±1.0) × k

The components of vector c that are perpendicular to vector a, are in the xy plane and have a magnitude of 5.0 units are (±4.9, 0, ±1.0), where the sign is chosen according to the right-hand rule.

Therefore, using the given information, we found the angle between vectors a and b to be approximately 69.7 degrees. Additionally, we found the components of a vector c that is perpendicular to a, is in the xy plane and has a magnitude of 5.0 units to be (±4.9, 0, ±1.0).

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To find the time it takes the rocket to reach a height of 900 feet,

The given formula is:

`h(t) = -16t² + 128t`.

substitute `900` for `h(t)` and solve for `t`. So, we have:

`h(t) = -16t² + 128t = 900`

Rearranging the above equation, we get the quadratic equation:

`16t² - 128t + 900 = 0`

We can solve this quadratic equation using factorizing, the quadratic formula, or completing the square. Here, we will use factoring. The factors of `16(900) = 14400` which add up to `-128` are `-100` and `-28`.So, we can write:

`16t² - 100t - 28t + 900 = 0`

Factoring the first two terms, and the last two terms, we get:

`(4t - 25)(4t + 36) = 0`

Setting each factor equal to zero, we get:

`4t - 25 = 0` or `4t + 36 = 0`

Solving for `t` in each equation, we get:

`t = 25/4` or `t = -9`

Since time can't be negative, we discard the negative solution.

Hence, the time it takes the rocket to reach a height of 900 feet is `25/4` seconds or `6.25` seconds (rounded to two decimal places).so the answer is: `6.25` seconds

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