   Chapter 6.2, Problem 15QY ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

#### Solutions

Chapter
Section ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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# The revenue (in millions of dollars) for a new product is modeled by R = 144 t 2 + 400 where t is the time in years.(a) Estimate the total revenue of the product over its first 3 years on the market.(b) Estimate the total revenue of the product over its first 6 years on the market.

(a)

To determine

To calculate: The total revenue of the product during first three years on market, when the revenue of product in millions of dollars is modelled as R=144t2+400.

Explanation

Given Information:

The revenue of product in millions of dollars is modelled as R=144t2+400.

Formula used:

The formula for integral u2±a2du is,

u2±a2du=12(uu2±a2±a2ln|u+u2±a2|)+C

Calculation:

Let u=12t.

Differentiate the above function with respect to x,

du=12dt

Consider the provided expression,

R=144t2+400

Integrate the above function with lower limit as 0 and higher limit as 3 in order to total revenue of the product during first three years on market, which can be written as,

R=03144t2+400dt

Rewrite.

R=03(12t)2+202dt

Multiply and divide by 12.

R=1120312(12t)2+202dt=11203((12t)2+202)12dt

Here, a=20, u=12t and du=12dt.

Substitute u for 12t, a for 20, and du for 12dt.

R=11203u2+a2du

Use the formula 23 and solve the above integral as:

R=112[12(uu2+a2+a2ln|u+u2+a2|)]03

Substitute 12t for u, and 20 for a

(b)

To determine

To calculate: The total revenue of the product during first six years on market, when the revenue of product in millions of dollars is modelled as R=144t2+400.

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