You may want to use the graph to help you solve the following problems.
Many application problems require that we find the maximum or minimum value for a function. Once we know the function defining the problem, the derivative allows us to find its maxium or minimum with relative ease. What follows are some interesting applications of this type.
In each of the following problems, verify that you have a maximum or minimum using the first or second derivative test.
You may want to use the graph to help you solve the following problems.
1. The estimated monthly profit for the Kodiak Camera Company is given by the function
where x is the number of cameras produced and sold and P is in dollars.
a. Determine the number of cameras Kodiak should produce and sell each month to maximize its profit.
b. Find the maximum profit.
2. The altitude , h (t ) (in feet) , obtained by a model rocket t seconds into its flight is given by the function
Verify all answers using calculus and algebra.
a. When is the rocket rising?
b. When is the rocket falling?
c. Find the maximum altitude obtained by the rocket.
d. Find the maximum velocity of the rocket.
3. Stonybrook Apartments owns 180 efficiencies which are fully occupied when the rent is $300 per month. Stonybrook estimates that for each $10 increase in rent, five apartments will become vacant. What rent should be charged so that Stonybrook obtains the largest gross income?
4. A Norman window has the shape of a rectangle surmounted by a semicircle .
a. If the perimeter of window is 28 feet, what should its dimensions be to allow the maximum amount of light through the window
b. What is the area?
5. At 1:00 P.M., ShipA is 30 nautical miles due south of Ship B and is sailing north at 15 knots per hour.
Recall: D = RT.
a. If Ship B is sailing west at a rate of 10 knots per hour, find the time at which the distance D between the ships is minimal.
b. How far apart are the ships?
6. The Denty Moore Company requires that its tin beef stew containers have a capacity of 54 cubic inches and the shape of a right circular cylinder. Determine the radius and height of the container that requires the least amount of material.
