   Chapter 10.2, Problem 35E ### Mathematical Applications for the ...

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

#### Solutions

Chapter
Section ### Mathematical Applications for the ...

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

# Production Suppose that the total number of units produced by a worker in t hours of an 8-hour shift can be modeled by the production function P(t): P ( t ) = 27 t + 12 t 2 − t 3 (a) Find the number of hours before production is maximized.(b) Find the number of hours before the rate of production is maximized. That is, find the point of diminishing returns.

(a)

To determine

To calculate: The number of hours before production is maximized where the production function is P(t)=27t+12t2t3, t hours of an 8 hours shift.

Explanation

Given Information:

The provided function is P(t)=27t+12t2t3.

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 P(t)=27t+12t2t3.

Calculate the first derivative of the function P(t)=27t+12t2t3.

P(t)=27+24t3t2

Set the first derivative equal to 0.

27+24t3t2=03(9+8tt2)=03(9t)(1+t)=0

Thus, either (9t)=0 or (1+t)=0.

First consider (9t)=0.

Solve for t.

9t=09=tt=9

Now consider (1+t)=0.

Solve for t.

1+t=01=tt=1

Thus, the critical points are t=9 and t=1

(b)

To determine

To calculate: The number of hours before rate of production is maximized where the production function is P(t)=27t+12t2t3, t hours of an 8 hours shift.

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