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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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A manufacturing company forecasts that the demand x (in units) for its product over the next 5 years can be modeled by

x = 1000 ( 45 + 20 t e 0.5 t ) ,   0 t 5 where t is the time in years.

(a) Find the total demand over the next 5 years.

(b) Find the average annual demand during the 5-year period.

(a)

To determine

To calculate: The total demand over next five years, when demand of the product over next five years is modelled as x=1000(45+20te0.5t) for 0t5.

Explanation

Given Information:

The demand of the product over next five years is modelled as x=1000(45+20te0.5t) for 0t5.

Formula used:

The formula for integration by parts is,

udv=uvvdu

Here, u and v be the differentiable functions x.

Integral Rule of exponential function:

enudu=enun+C

Here, n is the any number.

Calculation:

Consider the provided integral,

x=1000(45+20te0.5t)

Integrate the above function from lower limit as 0 and higher limit as 5 in order to find the total demand over next five years.

x=051000(45+20te0.5t)dt

Rewrite.

x=100005(45)dt+100005(20te0.5t)dt=4500005dt+2000005(te0.5t)dt

Let u=t and dv=e0.5tdt. So,

du=dt

And,

v=dv=e0.5tdt=e0.5t0.5=2e0.5t

Apply integration by parts and power rule of integration to solve integral of 4500005dt+2000005(te0

(b)

To determine

To calculate: The average annual demand during five years, when demand of the product over next five years is modelled as x=1000(45+20te0.5t) for 0t5.

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