Chapter 12.6, Problem 14P

To determine

To find speed in miles per hour by giving irrational roots to the nearest tenth.

Answer to Problem 14P

Kelly: around $10.2$ miles per hour.

Judith: around $12.2$ miles per hour.

Explanation of Solution

Given:

Judith can ride her bike $2\text{ mi/h}$ faster than Kelly. Judith takes $48\text{ min}$ min less to travel $50$ miles than Kelly does.

Calculation:

Let the Kelly’s speed be $x$ mi/hour, then Judith’s speed by $x+2$ mi/hour.

Now according to the question-

Judith’s time $=$ Kelly’s time $-\frac{48}{60}$

  $\begin{array}{l}\frac{50}{x+2}=\frac{50}{x}-\frac{4}{5}\\ \Rightarrow \frac{50}{x}-\frac{50}{x+2}=\frac{4}{5}\\ \Rightarrow \frac{50(x+2-x)}{(x)(x+2)}=\frac{4}{5}\\ \Rightarrow \frac{100}{x^2+2x}=\frac{4}{5}\\ \Rightarrow x^2+2x-125=0\\ \Rightarrow x^2+2x+1=125+1\{\text{completing the square}}\\ $ \Rightarrow {(x+1)}^2=126$\Rightarrow x+1=\sqrt{126}\\ \Rightarrow x_1=-1-\sqrt{126}=-12.2\\ \Rightarrow x_2=-1+\sqrt{126}=10.2\end{array}$

Therefore their speeds are:

Kelly: around $10.2$ miles per hour.

Judith: around $12.2$ miles per hour.