Problem Name

Problem Description

Notes

business_calculus_21

A publishing company sells 400,000 copies of a certain book each year. Ordering the entire amount printed at the beginning of the year ties up valuable storage space and capital. However, running off the copies in several partial runs throughout the year results in added costs for setting up each printing run. Setting up each production run costs $1000. The carrying costs, figured on the average number of books in storage, are 50 cents per book. Find the economic lot size, that is, the production run size that minimizes the total setting up and carrying costs.

business_calculus_22

The demand equation for a product is . Find the value of and the corresponding price that maximizes revenue.

business_calculus_23

A store estimates that its cost when selling lamps per day is dollars, where (i.e., the marginal cost per lamp is $40). Suppose that daily sales are rising at the rate of three lamps per day. How fast are the costs rising?

business_calculus_24

A small orchard yields 25 bushels of fruit per tree when planted with 40 trees. Because of overcrowding, the yield per tree (for each tree in the orchard) is reduced by 1⁄2 bushel for each additional tree that is planted. How many trees should be planted in order to maximize the total yield of the orchard?

business_calculus_25

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.

business_calculus_26

Let be the revenue received from the sale of units of a product. The average revenue per unit is defined by . Show that at the level of production where the average revenue is maximized, the average revenue equals the marginal revenue.

business_calculus_27

The weight of a male hognose snake is approximately grams, where is its length in meters. Suppose a snake has a length meters and is growing at the rate of 0.2 meters per year. At what rate is the snake gaining weight?

business_calculus_28

The volume of a spherical cancer tumor is given by , where is the diameter of the tumor. A physician estimates that the diameter is growing at the rate of 0.4 millimeters per day, at a time when the diameter is already 10 millimeters. How fast is the volume of the tumor changing at that time?

business_calculus_29

A company pays dollars in taxes when its annual profit is dollars. Suppose that is some (differentiable) function of and is some function of time . Give a chain rule formula for the time rate of change of taxes.

business_calculus_30

An offshore oil well is leaking oil onto the ocean surface, forming a circular oil slick about 0.005 meters thick. If the radius of the slick is meters, then the volume of oil spilled is cubic meters. Suppose that the oil is leaking at a constant rate of 20 cubic meters per hour, so that . Find the rate at which the radius of the oil slick is increasing, at a time when the radius is 50 meters.