   Chapter 11.3, Problem 77E 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

Revenue The monthly sales of Sunny Electronics’ new sound system are given by q ( t ) = 2 , 000 t − 100 t 2 units per month, t months after its introduction. The price Sunny charges is p ( t ) = 1 , 000 − t 2 dollars per sound system, t months after introduction. Find the rate of change of monthly sales, the rate of change of the price, and the rate of change of monthly revenue 5 months after the introduction of the sound system. Interpret your answer. [HINT: See Example 4(a).]

To determine

To calculate: The rate of change of monthly sales, prices and a monthly revenue of Sunny Electronics’ after 5 months where the monthly sales are given by the function q(t)=2,000t100t2 units per month and prices of one sound system is given by the function p(t)=1,000t2 dollars and t is in months. Interpret the answers.

Explanation

Given Information:

The monthly sales are given by the function q(t)=2,000t100t2 units per month and prices of one sound system is given by the function p(t)=1,000t2 dollars and t is in months.

Formula used:

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)

A derivative of a constant is 0.

A derivative of a function y=xn using power rule is dydx=nxn1.

Difference rule of the derivative is ddx[f(x)g(x)]=ddx[f(x)]ddx[g(x)] where f(x) and g(x) are any two differentiable functions.

Constant multiple rules of derivative of a function f(x) is ddx[cf(x)]=cddx[f(x)] where c is constant.

Calculation:

Consider the function q(t)=2,000t100t2 and p(t)=1,000t2,

Calculate the derivative of the function q(t) with respect to t.

Apply difference rule and constant multiple rules,

q'(t)=2,000ddt(t)100ddt(t2)

Apply power rule,

q'(t)=2,000(1t11)100(2t21)=2000200t

So, the rate of change in sales is q'(t)=2000200t.

Calculate the value of the derivative q'(t)=2000200t at t=5,

q'(5)=2000200(5)=20001000=1000

Thus, the rate of increase of sales is 1000 units per month.

Calculate the derivative of the function p(t) with respect to t.

Apply difference rule and constant multiple rules,

p'(t)=1,000ddt(1)ddt(t2)

Apply power rule and constant derivative rule,

p'(t)=1000(0)2t21=2t

So, the rate of change of price is p'(t)=2t.

Calculate the value of the derivative p'(t)=2t at t=5,

p'(5)=2(5)=10

Thus, the rate of decrease in the prices is \$10 per month

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