# The speed of the woman ### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071 ### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

#### Solutions

Chapter 1, Problem 83RE
To determine

## To calculate: The speed of the woman

Expert Solution

Her running speed is 3.78 mile/hour.

### Explanation of Solution

Given information:

The distance she covers by cycling = 4 miles

The distance she covers by running= 2.5 miles

Formula used:

First of all, we want to know her running speed. Let's call that x

x = running speed (miles / hour)

Let's call her cycling speed y...

y = cycling speed (miles / hour)

So, she cycles for 4 miles, and runs for 2.5 miles

4 miles ÷y miles / hour is how long it takes her to cycle for 4 miles

2.5 miles ÷x miles / hour is how long it takes her to run for 2.5 miles

Since this whole process takes 1 hour:

4y+2.5x=1

Get rid of those fractions by multiplying both sides by (x)(y)

4x+2.5y=xy

Calculation:

From the question, we know that her cycling speed is 8 miles / hour faster than her running speed. So, we can say that

y=x+8

Replace y in our equation, then:

4x+2.5(x+8)=x(x+8)4x+2.5x+20=x2+8x

Combine like terms:

20=x2+1.5x

And get this into the form of a quadratic equation:

x2+1.5x20=0

Plug the numbers into the quadratic formula ,

Where  a=1,b=1.5 andc=20

From that, we get

x=3.78

OR

x=5.28

As this woman cannot run -5.28 miles per hour ( negative speed), so her running speed (x) must be 3.78 miles/hour, and her cycling speed (8 miles/hour faster) is 11.78 miles/hour

Verification:

2.5 miles, at 3.78 miles/hour would take 0.661 hours

4 miles, at 11.78 miles/hour would take 0.339 hours

For a total of 1 hour!

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