   Chapter 10.1, Problem 51E ### 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

# Productivity A time study showed that, on average, the productivity of a worker after t hours on the job can be modeled by P ( t ) = 27 t + 6 t 2 − t 3 ,  0 ≤ t ≤ 8 where P is the number of units produced per hour.(a) Find the critical values of this function.(b) Which critical value makes sense in this model?(c) For what values of t is P increasing?(d) Graph the function for 0 ≤ t ≤ 8 .

(a)

To determine

To calculate: The critical values of the function P(t)=27t+6t2t3,0t8 if a time study showed that on an average, the productivity of a worker on the job can be modeled by the function after t hours.

Explanation

Given Information:

The provided function is P(t)=27t+6t2t3,0t8.

Formula Used:

The critical values are the only values at which the graph can have turning points, the derivative cannot change sign anywhere except at the critical value.

This, in an interval between two critical values, the sign of the derivative at any value in the interval will be the sign of the derivative at all values in the interval.

As per the First Derivative Test,

The first derivative of the function is evaluated. The first derivative is made equal to zero in order to get the critical points.

The values of the critical values are kept inside the original function which gives the critical points. The intervals of the values of x are then evaluated for the relative maximum and minimum.

If the quadratic equation is of the form ax2+bx+c=0, then the roots will be equal to:

b±b24ac2a

Calculation:

Consider the provided equation P(t)=27t+6t2t3,0t8,

The critical values are the only values at which the graph can have turning points, the derivative cannot change sign anywhere except at the critical value.

Hence, there will no change in the values of critical values as in the derivative graph

(b)

To determine

The sense of the critical values of the function P(t)=27t+6t2t3,0t8 if a time study showed that on an average, the productivity of a worker on the job can be modeled by the function after t hours.

(c)

To determine

To calculate: The values of t for which the function P(t)=27t+6t2t3,0t8 is increasing if a time study showed that on an average, the productivity of a worker on the job can be modeled by the function after t hours.

(d)

To determine

To graph: The function P(t)=27t+6t2t3,0t8 is increasing if a time study showed that on an average, the productivity of a worker on the job can be modeled by the function after t hours.

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