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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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Sales The cumulative sales S (in thousands of units) of a new product after it has been on the market for t years are modeled by

S =   C e k / t .

During the first year, 5000 units were sold. The saturation point for the market is 30,000 units. That is, the limit of S as t . ) is 30,000.

(a) Solve for C and k in the model.

(b) How many units will be sold after 5 years?

(c) Use a graphing utility to graph the sales function.

(a).

To determine

To calculate: The value of C and k in model S=Cekt. Where model represents the sales S (in thousands of units) of a new product after, it is in the market for t number of years. If in the first year, the units of new product sold in the market were 5000 units. The saturation point of the product for the market is 30000 units, which means the limit of S as t is 30000.

Explanation

Given Information:

The provided sales model is S=Cekt and in the first year, the units of new product sold in the market were 5000 units. The saturation point of the product for the market is 30000 units.

Calculation:

Consider the sales equation S=Cekt.

The sale for the first year is 5000 units, means when t=1, S=5.

Substituting the values of S and t in the provided model,

S=CektCek=5

Now, sales is 30 units when t.

Substituting the values,

S=limtCektlimtCek=30

Apply the limit as t

limtCek=30C=30

Substitute the value of C=30 in Cek=5

(b)

To determine

The number of units that will sold after 5 years by using the model S=Cekt. Where model represents the sales S (in thousands of units) of a new product after, it is in the market for t number of years. If in the first year, the units of new product sold in the market was 5000 unit. The saturation point of the product for the market is 30000 units, which means the limit of S as t is 30000.

(c)

To determine

To graph: The model S=Cekt using the graphing calculator. Where model represents the sales S (in thousands of units) of a new product after, it is in the market for t number of years. If in the first year, the units of new product sold in the market was 5000 unit.

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