Chapter 1, Problem 83RE

To determine

To calculate: The speed of the woman

Answer to Problem 83RE

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:

  $\begin{array}{l}4x+2.5(x+8)=x(x+8)\\ 4x+2.5x+20=x2+8x\end{array}$

Combine like terms:

  $20=x2+1.5x$

And get this into the form of a quadratic equation:

  $x2+1.5x-20=0$

Plug the numbers into the quadratic formula ,

Where  $a=1,b=1.5andc=-20$

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!