   Chapter 10.4, Problem 6E ### 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   =   27 t   +   6 t 2 — t 3 ,   0 ≤ t ≤ 8 where P is the number of units produced per hour. After how many hours will productivity be maximized? What is the maximum productivity?

To determine

To calculate: The number of hours to maximize the productivity and the maximum productivity.

Explanation

Given Information:

The number of units produced per hour is given as:

P=27t+6t2t3, 0t8

Where t denotes the hours.

Formula used:

To find the maximum value, calculate the relevant stationary value of an equation. Differentiate the function with respect to the independent variable and equate it to 0. the relevant value is the stationary value that satisfies the provided conditions.

If the second derivative of the provided function is less than zero, then substituting the value of the independent variable will give the maximum value of the equation.

The power rule is used for a function in which the expression can be written as every term raised to a power (be it fractional, positive or negative). For the function f(x)=xn, the derivative is

ddx[xn]=nxn1

Formula to find the roots of a quadratic equation ax2+bx+c=0 is given as

x=b±b24ac2a

Calculation:

As given, the number of units produced per hour is given by the equation:

P=27t+6t2t3, 0t8

Where t denotes the hours.

To find the stationary value for P=27t+6t2t3, differentiate the equation with respect to the independent variable t to get

P'=27+6×2t3t2=27+12t3t2

Put P'=0 to get t,

3t2+12t+27

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