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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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Revenue A company sells a seasonal product. The revenue R (in dollars) generated by sales of the product can be modeled by

R = 410.5 t 2 e r / 30 + 25 , 000 ,  0 t 365 where t is the time in days.

(a) Find the average daily revenue during the first quarter, which is given by 0   t   90 .

(b) Find the average daily revenue during the fourth quarter, which is given by 274   t   365 .

(c) Find the total daily revenue during the year.

(a)

To determine

To calculate: The average daily revenue during the first quarter given by 0t90 when the company’s revenue in dollars is generated by sales of the product is given by the function,

R=410.5t2et/30+25,000   0<t365

Where t is the time in days.

Explanation

Given Information:

The company’s revenue in dollars is generated by sales of seasonal product is given by the function,

R=410.5t2et/30+25,000   0<t365

Where t is the time in days.

Formula used:

Integration by parts.

When u and v is assumed to be the differentiable functions of x then,

u dv=uvv du

Also,

xneax dx=xneaxanaxn1eaxdx

Calculation:

The company’s revenue in dollars is given by the function,

R=410.5t2et/30+25,000   0<t365

In order to calculate company’s revenue during the first quarter given by 0t90 integrate the function R=410.5t2et/30+25,000 with lower limit as 0 and upper limit as 90.

Thus,

Average value=1901090(410.5t2et/30+25,000) dt=25000090dt+410.5090(t2et/30) dt

In order to obtain the value of the integral 410.5090(t2et/30) dt apply the formula xneax dx=xneaxanaxn1eaxdx two times, first with n=2,a=130, then with n=1,a=130

Average value=1900[25000090dt+410

(b)

To determine

To calculate: The average daily revenue during the fourth quarter given by 274t365 when the company’s revenue in dollars is generated by sales of the product is given by the function,

R=410.5t2et/30+25,000   0<t365

Where t is the time in days.

(c)

To determine

To calculate: The total daily revenue of company during the year when the company’s revenue in dollars is generated by sales of the product is given by the function,

R=410.5t2et/30+25,000   0<t365

Where t is the time in days.

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