91Ó°ÊÓ

The marginal cost of a product can be thought of as the cost of producing one additional unit of output. For example, if the marginal cost of producing the \(50 t h\) product is \(\$ 6.20,\) it costs \(\$ 6.20\) to increase production from 49 to 50 units of output. Suppose the marginal cost \(C\) (in dollars) to produce \(x\) thousand digital music players is given by the function $$ C(x)=x^{2}-140 x+7400 $$ (a) How many players should be produced to minimize the marginal cost? (b) What is the minimum marginal cost?

The monthly revenue \(R\) achieved by selling \(x\) wristwatches is \(R(x)=75 x-0.2 x^{2} .\) The monthly cost \(C\) of selling \(x\) wristwatches is $$ C(x)=32 x+1750 $$ (a) How many wristwatches must the firm sell to maximize revenue? What is the maximum revenue? (b) Profit is given as \(P(x)=R(x)-C(x)\). What is the profit function? (c) How many wristwatches must the firm sell to maximize profit? What is the maximum profit? (d) Provide a reasonable explanation as to why the answers found in parts (a) and (c) differ. Explain why a quadratic function is a reasonable model for revenue.

The daily revenue \(R\) achieved by selling \(x\) boxes of candy is \(R(x)=9.5 x-0.04 x^{2}\). The daily cost \(C\) of selling \(x\) boxes of candy is \(C(x)=1.25 x+250 .\) (a) How many boxes of candy must the firm sell to maximize revenue? What is the maximum revenue? (b) Profit is given as \(P(x)=R(x)-C(x) .\) What is the profit function? (c) How many boxes of candy must the firm sell to maximize profit? What is the maximum profit? (d) Provide a reasonable explanation as to why the answers found in parts (a) and (c) differ. Explain why a quadratic function is a reasonable model for revenue.

An accepted relationship between stopping distance \(d\) (in feet), and the speed \(v\) of a car (in \(\mathrm{mph}\) ), is \(d=1.1 v+0.06 v^{2}\) on dry, level concrete. (a) How many feet will it take a car traveling \(45 \mathrm{mph}\) to stop on dry, level concrete? (b) If an accident occurs 200 feet ahead of you, what is the maximum speed you can be traveling to avoid being involved?

On one set of coordinate axes, graph the family of \(\begin{array}{ll}\text { parabolas } f(x)=x^{2}+2 x+c \text { for } c=-3, & c=0 \text { , }\end{array}\) and \(c=1\). Describe the characteristics of a member of this family.

Why is the graph of a quadratic function concave up if \(a>0\) and concave down if \(a<0 ?\)