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Mathematical Applications for the ...

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

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BuyFindarrow_forward

Mathematical Applications for the ...

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

Advertising and sales—diminishing returns Suppose that a company’s daily sales volume attributed to an advertising campaign is given by

S ( t ) = 3 t + 3 18 ( t + 3 ) 2 + 1

(a) Find how long it will be before sales volume is maximized.

(b) Find how long it will be before the rate of change of sales volume is minimized. That is, find the point of diminishing returns.

(a)

To determine

To calculate: The number of days before sales is maximized where the sales function is S(t)=3t+318(t+3)2+1.

Explanation

Given Information:

The provided function is S(t)=3t+318(t+3)2+1.

Formula used:

To find relative maxima and minima of a function,

1. Set the first derivative of the function to zero, f'(x)=0, to find the critical values of the function.

2. Substitute the critical values into f(x) and calculate the critical points.

3. Calculate the sign of the function to the left and right of the critical values.

(a) If f(x)<0 and f(x)>0 on the left and right side respectively of a critical value, then the point is relative minimum.

(b) If f(x)>0 and f(x)<0 on the left and right side respectively of a critical value, then the point is relative maximum.

Calculation:

Consider the function S(t)=3t+318(t+3)2+1.

Calculate the first derivative of the function S(t)=3t+318(t+3)2+1.

S(t)=3(t+3)2+36(t+3)3=3(t+3)+36(t+3)3=273t(t+3)3

Set the first derivative equal to 0.

273t(t+3)3=0273t=0

Thus, 273t=0

(b)

To determine

To calculate: The number of days before rate of change of sales is minimized where the sales function is S(t)=3t+318(t+3)2+1.

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