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Chapter 1: Problem 59
Sketch the straight line defined by the linear equation by finding the \(x\) - and \(y\) -intercepts. $$ x+2 y-4=0 $$
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The United States is not building many nuclear plants, but the ones it has are running at nearly full capacity. The output (as a percent of total capacity) of nuclear plants is described by the equation $$ y=1.9467 t+70.082 $$ where \(t\) is measured in years, with \(t=0\) corresponding to the beginning of 1990 . a. Sketch the line with the given equation. b. What are the slope and the \(y\) -intercept of the line found in part (a)? c. Give an interpretation of the slope and the \(y\) -intercept of the line found in part (a). d. If the utilization of nuclear power continues to grow at the same rate and the total capacity of nuclear plants in the United States remains constant, by what year can the plants be expected to be generating at maximum capacity?
Suppose the cost function associated with a product is \(C(x)=c x+F\) dollars and the revenue function is \(R(x)=\) \(s x\), where \(c\) denotes the unit cost of production, \(s\) the unit selling price, \(F\) the fixed cost incurred by the firm, and \(x\) the level of production and sales. Find the break-even quantity and the break-even revenue in terms of the constants \(c, s\), and \(F\), and interpret your results in economic terms.
Moody's Corporation is the holding company for Moody's Investors Service, which has a \(40 \%\) share in the world credit-rating market. According to Company Reports, the total revenue (in billions of dollars) of the company is projected to be as follows \((x=0\) correspond to 2004\()\) : $$ \begin{array}{lccccc} \hline \text { Year } & 2004 & 2005 & 2006 & 2007 & 2008 \\ \hline \text { Revenue, } \boldsymbol{y} & 1.42 & 1.73 & 1.98 & 2.32 & 2.65 \\\ \hline \end{array} $$ a. Find an equation of the least-squares line for these data. b. Use the results of part (a) to estimate the rate of change of the revenue of the company for the period in question. c. Use the result of part (a) to estimate the total revenue of the company in 2010 , assuming that the trend continues.
For each demand equation, where \(x\) represents the quantity demanded in units of 1000 and \(p\) is the unit price in dollars, (a) sketch the demand curve and (b) determine the quantity demanded corresponding to the given unit price \(p\). $$ p=-3 x+60 ; p=30 $$
Suppose the demand-and-supply equations for a certain commodity are given by \(p=a x+b\) and \(p=c x+d\), respectively, where \(a<0, c>0\), and \(b>d>0\) (see the accompanying figure). a. Find the equilibrium quantity and equilibrium price in terms of \(a, b, c\), and \(d\). b. Use part (a) to determine what happens to the market equilibrium if \(c\) is increased while \(a, b\), and \(d\) remain fixed. Interpret your answer in economic terms. \(\mathbf{c}\). Use part (a) to determine what happens to the market equilibrium if \(b\) is decreased while \(a, c\), and \(d\) remain fixed. Interpret your answer in economic terms.
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