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Chapter 2: Problem 42
What does it mean if two quantities vary inversely?
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Among all pairs of numbers whose difference is \(24,\) find a pair whose product is as small as possible. What is the minimum product?
Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The graph of a rational function can never cross a vertical asymptote.
Among all pairs of numbers whose sum is \(20,\) find a pair whose product is as large as possible. What is the maximum product?
Among all pairs of numbers whose difference is \(16,\) find a pair whose product is as small as possible. What is the minimum product?
During the 1980 s, the controversial economist Arthur Laffer promoted the idea that tax increases lead to a reduction in government revenue. Called supply- side economics, the theory uses functions such as $$f(x)=\frac{80 x-8000}{x-110}, 30 \leq x \leq 100$$ This function models the government tax revenue, \(f(x),\) in tens of billions of dollars, in terms of the tax rate, \(x\). The graph of the function is shown. It illustrates tax revenue decreasing quite dramatically as the tax rate increases. At a tax rate of (gasp) \(100 \%\), the government takes all our money and no one has an incentive to work. With no income earned, zero dollars in tax revenue is generated. (GRAPH CAN'T COPY). During the 1980 s, the controversial economist Arthur Laffer promoted the idea that tax increases lead to a reduction in government revenue. Called supply- side economics, the theory uses functions such as $$f(x)=\frac{80 x-8000}{x-110}, 30 \leq x \leq 100$$ This function models the government tax revenue, \(f(x),\) in tens of billions of dollars, in terms of the tax rate, \(x\). The graph of the function is shown. It illustrates tax revenue decreasing quite dramatically as the tax rate increases. At a tax rate of (gasp) \(100 \%\), the government takes all our money and no one has an incentive to work. With no income earned, zero dollars in tax revenue is generated. (GRAPH CAN'T COPY). Use function \(f\) and its graph to solve Exercises \(55-56\) a. Find and interpret \(f(30)\). Identify the solution as a point on the graph of the function. b. Rewrite the function by using long division to perform $$(80 x-8000) \div(x-110)$$ Then use this new form of the function to find \(f(30) .\) Do you obtain the same answer as you did in part (a)? c. Is \(f\) a polynomial function? Explain your answer.
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