/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 118 See Exercise \(117 .\) Suppose a... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 118

See Exercise \(117 .\) Suppose an economist determines that $$R(x)=\frac{60 x-6000}{x-120}$$ where \(y=R(x)\) is government revenue in tens of millions of dollars for a tax rate of \(x\) percent, with \(y=R(x)\) valid for \(50 \leq x \leq 100 .\) Find the revenue for each tax rate. (a) \(50 \%\) (b) \(60 \%\) (c) \(80 \%\) (d) \(100 \%\) (e) Graph \(R\) in the window \([0,100]\) by \([0,50]\)

Short Answer

Expert verified
The revenues are approximately 42.857, 40, 30, and 0 tens of millions of dollars for the tax rates of 50%, 60%, 80%, and 100% respectively.

Step by step solution

01

Identify the given function

The function given is the revenue function: \[ R(x) = \frac{60x - 6000}{x - 120} \] where \( R(x) \) is the government revenue in tens of millions of dollars and \( x \) is the tax rate in percent.
02

Compute the revenue at 50%

Substitute \( x = 50 \) into the function: \[ R(50) = \frac{60(50) - 6000}{50 - 120} = \frac{3000 - 6000}{-70} = \frac{-3000}{-70} = 42.857 \] So, the revenue at 50% is approximately 42.857 tens of millions of dollars.
03

Compute the revenue at 60%

Substitute \( x = 60 \) into the function: \[ R(60) = \frac{60(60) - 6000}{60 - 120} = \frac{3600 - 6000}{-60} = \frac{-2400}{-60} = 40 \] So, the revenue at 60% is 40 tens of millions of dollars.
04

Compute the revenue at 80%

Substitute \( x = 80 \) into the function: \[ R(80) = \frac{60(80) - 6000}{80 - 120} = \frac{4800 - 6000}{-40} = \frac{-1200}{-40} = 30 \] So, the revenue at 80% is 30 tens of millions of dollars.
05

Compute the revenue at 100%

Substitute \( x = 100 \) into the function: \[ R(100) = \frac{60(100) - 6000}{100 - 120} = \frac{6000 - 6000}{-20} = \frac{0}{-20} = 0 \] So, the revenue at 100% is 0 tens of millions of dollars.
06

Graph the function

To graph the function, plot the points from above on the coordinate plane in the window \([0, 100]\) by \([0, 50]\). Plot the following points for greater accuracy: - (50, 42.857) - (60, 40) - (80, 30) - (100, 0)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rational Functions
Rational functions are fractions where both the numerator and the denominator are polynomials. In this exercise, the function given is a rational function: \( R(x) = \frac{60x - 6000}{x - 120} \). The numerator here is \(60x - 6000 \) and the denominator is \( x - 120 \). Rational functions can have vertical asymptotes鈥攍ines where the function goes to infinity. In this case, there's a vertical asymptote at \( x = 120 \), because the denominator becomes zero and the function is undefined at this point. Understanding the behavior near these special points helps in sketching the overall graph.
Graphing Functions
Graphing a function helps visualize its behavior over different values. Here, the given function \( R(x) = \frac{60x - 6000}{x - 120} \) should be graphed within the window \[0, 100\] by \[0, 50\]. Start by finding key points. For example:
  • At \( x = 50\), \( R(50) = 42.857 \)
  • At \( x = 60\), \( R(60) = 40 \)
  • At \( x = 80\), \( R(80) = 30 \)
  • At \( x = 100\), \( R(100) = 0 \)
These points can be plotted on the coordinate plane. Then, sketch the curve through these points, noticing the sharp drop because of the asymptote at \( x = 120 \). Make sure to smoothly connect the dots and observe how the curve behaves as \( x \) approaches 120 from both sides.
Tax Rate Analysis
Tax rate analysis involves understanding how taxes affect government revenue. Here, the function \( R(x) \) shows how different tax rates \( x \) yield different revenue values \( R(x) \). By evaluating the function at various tax rates, the government can see:
  • For a 50% tax rate, the revenue is approximately 42.857 tens of millions of dollars.
  • For a 60% tax rate, it is 40 tens of millions of dollars.
  • At 80% tax rate, it is 30 tens of millions of dollars.
  • At 100% tax rate, the revenue drops to 0 tens of millions of dollars.
Notice how increasing the tax rate initially increases revenue, but raising it further decreases the revenue because it eventually dissuades taxable activities. This is a classical representation of the Laffer Curve, which demonstrates the balance needed in setting tax rates to optimize revenue.

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Most popular questions from this chapter

Match the rational function in Column I with the appropriate description in Column II. Choices in Column II can be used only once. A. The \(x\) -intercept is \(-3\) B. The \(y\) -intercept is 5 C. The horizontal asymptote is \(y=4\) D. The vertical asymptote is \(x=-1\) E. There is a "hole" in its graph at \(x=-4\) F. The graph has an oblique asymptote. G. The \(x\) -axis is its horizontal asymptote, and the \(y\) -axis is not its vertical asymptote. H. The \(x\) -axis is its horizontal asymptote, and the \(y\) -axis is its vertical asymptote. $$f(x)=\frac{x+3}{x-6}$$

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Solve each problem. Sum and Product of Two Numbers Find two numbers whose sum is 20 and whose product is the maximum possible value. (Hint: Let \(x\) be one number. Then \(20-x\) is the other number. Form a quadratic function by multiplying them, and then find the maximum value of the function.

Queuing theory (also known as waiting-line theory) investigates the problem of providing adequate service economically to customers waiting in line. Suppose customers arrive at a fast-food service window at the rate of 9 people per hour. With reasonable assumptions, the average time (in hours) that a customer will wait in line before being served is modeled by $$f(x)=\frac{9}{x(x-9)}$$ where \(x\) is the average number of people served per hour. A graph of \(f(x)\) for \(x>9\) is shown in the figure on the next page. (a) Why is the function meaningless if the average number of people served per hour is less than \(9 ?\) Suppose the average time to serve a customer is 5 min. (b) How many customers can be served in an hour? (c) How many minutes will a customer have to wait in line (on the average)? (d) Suppose we want to halve the average waiting time to \(7.5 \mathrm{min}\left(\frac{1}{8} \mathrm{hr}\right) .\) How fast must an employee work to serve a customer (on the average)? (Hint: Let \(f(x)=\frac{1}{8}\) and solve the equation for \(x .\) Convert your answer to minutes.) How might this reduction in serving time be accomplished?
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