   Chapter 9, Problem 28RE

Chapter
Section
Textbook Problem

# Elvish for Dummies The sales department at OHaganBooks.com predicts that the revenue from sales of the latest block- buster Elvish for Dummies will vary in accordance with annual releases of episodes of the movie series “Lord of the Rings Episodes 9–12.” It has come up with the following model (which includes the effect of diminishing sales): R ( t ) = 20 , 000 + 15 , 000 e − 0.12 t cos [ π 6 ( t − 4 ) ] dollars                                                                      ( 0 ≤ t ≤ 72 ) , where t is time in months from now and R ( t ) is the monthly revenue. How fast, to the nearest dollar, will the revenue be changing 10 months from now?

To determine

To calculate: The rate at which revenue to the nearest dollar be changing 10 months from now for the sales department of OHaganBooks.com which predicts that the revenue from sales of the latest blockbuster Elvish for Dummies will vary in accordance with annual releases of episodes of the movie series “Lord of the Rings Episodes 912 ”. It has come up with the model R(t)=20,000+15,000e0.12tcos[π6(t4)] dollars 0t72, where t is the time in months from now and R(t) is the monthly revenue.

Explanation

Given Information:

The sales department of OHaganBooks.com which predicts that the revenue from sales of the latest blockbuster Elvish for Dummies will vary in accordance with annual releases of episodes of the movie series “Lord of the Rings Episodes 912 ”. It has come up with the model R(t)=20,000+15,000e0.12tcos[π6(t4)] dollars 0t72, where t is the time in months from now and R(t) is the monthly revenue.

Formula Used:

If f(x) and g(x) are two functions then sum and difference rules for the differentiation is,

ddx[f(x)±g(x)]=ddxf(x)±ddxg(x)

If f is a differentiable function of u and v of function x, then the product rule is given by:

ddxf(u)g(u)=g(u)ddxf(u)+f(u)dudxg(u)

Constant multiple rules are ddxkg(x)=kddxg(x) where k is a constant.

Calculation:

Consider the function R(t)=20,000+15,000e0.12tcos[π6(t4)].

Now evaluate the derivative of the function R(t)=20,000+15,000e0.12tcos[π6(t4)] where f(u)=20,000 and g(u)=15,000e0.12tcos[π6(t4)],

Apply the sum rule,

R(t)=ddt20,000+ddt15,000e0.12tcos[π6(t4)]

Now differentiate the constant term with respect to t,

R(t)=ddt20,000R(t)=0

Then apply the constant multiple rules and product rule to evaluate the differentiation of the function 15,000e0.12tcos[π6(t4)].

15,000e0.12tcos[π6(t4)]=15,000[ddte0.12tcos[π6(t4)]]=15,000[cos[π6(t4)]ddte0.12t+e0.12tddtcos[π6(t4)]]

Further solved as:

15,000e0

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