   Chapter 11, Problem 72RE Finite Mathematics and Applied Cal...

7th Edition
Stefan Waner + 1 other
ISBN: 9781337274203

Solutions

Chapter
Section Finite Mathematics and Applied Cal...

7th Edition
Stefan Waner + 1 other
ISBN: 9781337274203
Textbook Problem

Demand and Revenue The price p that OHaganBooks.com charges for its latest leather-bound gift edition of Lord of the Fields is related to the demand q in weekly sales by the equation 100 p q + q 2 = 5 , 000 , 000. Suppose the price is set at $40, which would make the demand 1,000 copies per week.a. Using implicit differentiation, compute the rate of change of demand with respect to price, and interpret the result. (Round the answer to two decimal places.)b. Use the result of part (a) to compute the rate of change of revenue with respect to price. Should the price be raised or lowered to increase revenue? (a) To determine To calculate: The rate of demand function change with price using the implicit differentiation and determine the significance of the result, if the demand q of the edition of “Lord of the Fields” in a week is copied its price p is$40 and the prices p related to the demand q is given by the equation 100pq+q2=5,000,000.

Explanation

Given Information:

The demand q of the edition of “Lord of the Fields” in a week is copied its price p is $40 and the prices p related to the demand q is given by the equation 100pq+q2=5,000,000. Formula used: Constant multiple rules of derivative of a function f(x) are f'(cx)=cf'(x) where c is constant. Product rule of derivative of differentiable functions, f(x) and g(x) is ddx[f(x)g(x)]=f(x)g(x)+f(x)g(x). The derivative of a function f(x)=un using the chain rule is f(x)=ddx(un)=nun1dudx where u is the function of x. Calculation: Consider the prices p related to the demand q, 100pq+q2=5,000,000 Find the rate of demand function with respect to price by determining the derivative. Take ddp of both sides of the equation, ddp(100pq+q2)=ddp(5,000,000)ddp(100pq)+ddp(q2)=0 Apply constant multiple rules, 100ddp(pq)+ddp(q2)=0 Apply the product rule and chain rule of derivative, 100(dpdpq+pdqdp (b) To determine To calculate: The rate of revenue with price using the implicit differentiation and determine whether the price increases or decrease to increase revenue, if the demand q of the edition of “Lord of the Fields” in a week is copied its price p is$40 and the prices p related to the demand q is given by the equation 100pq+q2=5,000,000.

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