Problem Name

Problem Description

Notes

diff_apps_1

The radius r of a sphere is increasing at a rate of 2 inches per minute. Find the rate of change of the volume when r = 24 inches.

diff_apps_2

Find the slope of the curve at .

diff_apps_3

From a 36-inch-square piece of cardboard, square corners are cut as shown in the diagram, and the resulting flaps are folded up to form an open box. Find the size of the squares to be cut from the corners so that the volume of the resulting box is maximized.

diff_apps_4

The weekly revenue for a product is given by

and the weekly cost is

where is the number of units produced and sold.

diff_apps_5

Find two positive numbers, and , whose product is 100 and whose sum is as small as possible.

diff_apps_6

A large cylindrically-shaped soup can is to be designed so that the can will hold cubic inches (about 28 ounces) of soup. Find the radius and height of the can for which the amount of metal needed is as small as possible.

diff_apps_7

The daily output of a coal mine after hours of operation is approximately tons, . Find the maximum rate of output (in tons of coal per hour).

diff_apps_8

In the planning of a sidewalk café, it is estimated that if there are 12 tables, the daily profit will be $10 per table. Because of overcrowding, for each additional table the profit per table (for every table in the café) will be reduced by $0.50. How many tables should be provided to maximize the profit from the café?

diff_apps_9

A closed rectangular box with a square base is to be constructed using two different types of wood. The top is made of wood costing $3 per square foot and the remainder is made of wood costing $1 per square foot. Suppose that $48 is available to spend. Find the dimensions of the box of greatest volume that can be constructed.

diff_apps_10

Suppose that a total cost function is . Is the marginal cost increasing, decreasing, or not changing when ? Find the minimum marginal cost.