Chapter 13.2, Problem 68E

### Mathematical Applications for the ...

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

Chapter
Section

### Mathematical Applications for the ...

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

# Product life The useful life of a car battery t (in years) is given by the probability density function f ( t )   =   0.2 e − 0.2 t ,     t ≥ 0 Find the probability that a battery chosen at random lasts(a) 2 years or less. (b) between 4 and 6 years.(c) more than 4 years given that it lasts less than 6 years.

(a)

To determine

To calculate: The probability that a battery chosen at random lasts 2 years or less if the useful life of a car battery where time t in years is given by the probability density function f(t)=0.2e0.2t, t0 .

Explanation

Given Information:

The probability density function f(t)=0.2e0.2t.

Formula used:

To calculate a definite integral, evaluate the indefinite integral and then substitute the limits of the integral.

The value of the integral eaxdx=eaxa where a is constant and x is variable.

Calculation:

Consider the probability density function f(t)=0.2e0.2t.

Since, a battery chosen at random lasts 2 years or less probability is the definite integral from 0 to 2.

Thus, probability that the battery takes 3 years or less is the integral 02(0.2e0.2t)dt.

Recall that to calculate a definite integral, evaluate the indefinite integral and then substitute the limits of the integral.

Consider the expression (0.2t).

Differentiate with respect to t.

d(0.2t)dt=0.2d(0.2t)=0.2dt

Thus, d(0.2t)=0.2dt.

Simplify the integral using d(0.2t)=0.2dt.

02(0

(b)

To determine

To calculate: The probability that a battery chosen at random lasts between 4 and 6 years if the useful life of a car battery where time t in years is given by the probability density function f(t)=0.2e0.2t, t0 .

(c)

To determine

To calculate: The probability that a battery chosen at random lasts more than 4 years given that it lasts less than 6 years if the useful life of a car battery where time t in years is given by the probability density function f(t)=0.2e0.2t, t0 .

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