   Chapter 10.5, Problem 37E ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

#### Solutions

Chapter
Section ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# Revenue A recently released film has its weekly revenue given by R ( t ) = 50 t t 2 + 36 ' t ≥ 0 where R(t) is in millions of dollars and t is in weeks.(a) Graph R(t).(b) When will revenue be maximized?(c) Suppose that if revenue decreases for 4 consecutive weeks, the film will be removed from theaters and will be released as a video 12 weeks later. When will the video come out?

(a)

To determine

To graph: The revenue function R(t)=50tt2+36.

Explanation

Given Information:

The provided function R(t)=50tt2+36, where t is the time in weeks and R(t) is revenue.

Graph:

Consider the provided function,

R(t)=50tt2+36

First calculate the rate of change of R(t) with respect to t.

Thus, differentiate the function with respect to t.

ddt(R(t))=ddt(50tt2+36)

Use the quotient rule to differentiate the function,

ddt(R(t))=ddt(50tt2+36)=(t2+36)ddt(50t)(50t)ddt(t2+36)(t2+36)2=(t2+36)(50)(50t)(2t)(t2+36)2=50t2+1800100t2(t2+36)2

Calculate further,

ddt(R(t))=180050t2(t2+36)2

The maximum exists at the critical value where R(t)=0

Thus, for critical value

0=180050t2(t2+36)20=180050t250t2=1800t2=36

Solve further,

t=6

Find R(7) to get the trend of the curve after the critical value t=6

R(7)=180050(7)2((7)2+36)2=180050(49)((7)2+36)2=18002450((7)2+36)2=650((7)2+36)2

Which is

R(7)<0

Therefore, the revenue starts decreasing after t=6

(b)

To determine

To calculate: The value of t at which R(t) in maximized for the function R(t)=50tt2+36 here t is the time in weeks and R(t) is revenue.

(c)

To determine

To calculate: The value of t at which the movie released as a video if it was to supposed to be removed from cinema after revenue decreases for 4 weeks and was supposed to be released as a video after 12 weeks of getting removed from cinema.

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