91Ó°ÊÓ

The weekly sales of Honolulu Red Oranges is given by \(q=1,000-20 p\). Calculate the price elasticity of demand when the price is \(\$ 30\) per orange (yes, \(\$ 30\) per orange \(^{63}\) ). Interpret your answer. Also, calculate the price that gives a maximum weekly revenue, and find this maximum revenue. HINT [See Example 1.]

The radius of a circular puddle is growing at a rate of \(5 \mathrm{~cm} / \mathrm{s}\) a. How fast is its area growing at the instant when the radius is \(10 \mathrm{~cm}\) ? HINT [See Example 1.] b. How fast is the area growing at the instant when it equals \(36 \mathrm{~cm}^{2}\) ? HINT [Use the area formula to determine the radius at that instant.]

A general linear demand function has the form \(q=m p+b(m\) and \(b\) constants, \(m \neq 0\) ). a. Obtain a formula for the price elasticity of demand at a unit price of \(p\). b. Obtain a formula for the price that maximizes revenue.

Economist Henry Schultz devised the following demand function for corn: $$p=\frac{6,570,000}{q^{1.3}}$$ where \(q\) is the number of bushels of corn that could be sold at \(p\) dollars per bushel in one year. \({ }^{7}\) Assume that at least 10,000 bushels of corn per year must be sold. a. How much should farmers charge per bushel of corn to maximize annual revenue? b. How much corn can farmers sell per year at that price? c. What will be the farmers' resulting revenue?

A baseball diamond is a square with side \(90 \mathrm{ft}\). A batter at home base hits the ball and runs toward first base with a speed of \(24 \mathrm{ft} / \mathrm{s}\). At what rate is his distance from third base increasing when he is halfway to first base?

Your underground used-book business is booming. Your policy is to sell all used versions of Calculus and You at the same price (regardless of condition). When you set the price at $$\$ 10$$, sales amounted to 120 volumes during the first week of classes. The following semester, you set the price at \(\$ 30\) and sold not a single book. Assuming that the demand for books depends linearly on the price, what price gives you the maximum revenue, and what does that revenue amount to?

A company manufactures cylindrical metal drums with open tops with a volume of 1 cubic meter. What should be the dimensions of the drum in order to use the least amount of metal in their production? HINT [See Example 4.]

A right circular conical vessel is being filled with green industrial waste at a rate of 100 cubic meters per second. How fast is the level rising after \(200 \pi\) cubic meters have been poured in? The cone has a height of \(50 \mathrm{~m}\) and a radius of \(30 \mathrm{~m}\) at its brim. (The volume of a cone of height \(h\) and crosssectional radius \(r\) at its brim is given by \(V=\frac{1}{3} \pi r^{2} h .\).)

A company manufactures cylindrical metal drums with open tops with a volume of 2 cubic meters. The metal used to manufacture the cans costs $$\$ 2$$ per square meter for the sides and $$\$ 3$$ per square meter for the (thicker) bottom. What should be the dimensions of the drums in order to minimize the cost of metal in their production? What is the ratio height/radius? HINT [See Example 4.]

Chocolate Box Company is going to make open-topped boxes out of \(6 \times 16\) -inch rectangles of cardboard by cutting squares out of the corners and folding up the sides. What is the largest volume box it can make this way?