   Chapter 9.2, Problem 31E ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

#### Solutions

Chapter
Section ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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# Learning Theory The time t (in hours) required for a new employee to learn to successfully operate a machine in a manufacturing process is described by the probability density function f ( t ) = 5 324 t 9 − t , [ 0 , 9 ] .Find the probability that a new employee will learn to operate the machine (a) in less than 3 hours and (b) in more than 4 hours but less than 8 hours.

(a)

To determine

To calculate: The probability if a new employee is learning to operate the machine in less than 3 hours. If the probability density function is represented by, f(t)=5324t9t, [0,9], where t(in hours) represents the time required for a new employee to operate the machine successfully.

Explanation

Given Information:

The probability density function is represented by, f(t)=5324t9t, [0,9], where t(in hours) represents the time required for a new employee to operate the machine successfully.

Formula used:

If f is function of a continuous random variable t, then the probability that t lies in the interval [c,d] is,

P(ctd)=cdf(t) dt

Calculation:

Consider the provided probability density function,

f(t)=5324t9t, [0,9],

Since, a new employee is learning to operate the machine in less than 3 hours. therefore,

The interval in which probability lies is 0<t<3.

Now, apply the formula of probability P(ctd)=cdf(t) dt.

Substitute the values 0 for c, 3 for d and 5324t9t for f(t) in the above formula as,

P(0<t<3)=035324t9t dt

First solve the integral 5324t9t dt by substitution method,

Let u=9t then t=9u and, dt=du.

Now, substitute the above values in the integral 5324t9t dt.

Therefore,

5324t9t dt=5324(9u)u12(du)=5324(9u)u12(du)=5324(u329u12)du=5324(25u526u32)

Substitute (9t

(b)

To determine

To calculate: The probability if a new employee is learning to operate the machine in more than 4 hours but less than 8 hours. If the probability density function is represented by, f(t)=5324t9t, [0,9], where t(in hours) represents the time required for a new employee to operate the machine successfully.

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