Chapter 8.6, Problem 10CYU

a

To determine

To write an expression for Alicia’s time with the wind

Answer to Problem 10CYU

An expression for Alicia’s time with the wind is $\frac{20}{x+11.5}$

Explanation of Solution

Given information :

Alicia’s average speed riding her bikes is 11.5 miles per hours and she take round trip of 40 miles. She takes 1 hours and 20 minutes with the wind and 2 hours 30 minutes against the wind.

Let the speed of the wind be $x$

Therefore, when Alicia travels with the wind, her speed will become $11.5+x$

Therefore the round trip is 40 miles long so each way is $\frac{40}{2}=20\text{ miles}$ .

Therefore, an expression for Alicia’s time with the wind is

  $\frac{20}{x+11.5}$

Hence, the An expression for Alicia’s time with the wind is $\frac{20}{x+11.5}$ .

b

To determine

To write an expression for Alicia’s time against the wind

Answer to Problem 10CYU

An expression for Alicia’s time against the wind is $\frac{20}{11.5-x}$

Explanation of Solution

Given information :

Alicia’s average speed riding her bikes is 11.5 miles per hours and she take round trip of 40 miles. She takes 1 hours and 20 minutes with the wind and 2 hours 30 minutes against the wind.

Let the speed of the wind be $x$

Therefore, the round trip is 40 miles long so each way is $\frac{40}{2}=20\text{ miles}$ .

Therefore, an expression for Alicia’s time against the wind is

  $\frac{20}{11.5-x}$

Hence, an expression for Alicia’s time against the wind is $\frac{20}{11.5-x}$ .

c

To determine

To find the total time to take to complete the trip.

Answer to Problem 10CYU

The total time to take to complete the trip is 3 hours 50 minutes

Explanation of Solution

Given information :

Alicia’s average speed riding her bikes is 11.5 miles per hours and she take round trip of 40 miles. She takes 1 hours and 20 minutes with the wind and 2 hours 30 minutes against the wind.

Let the speed of the wind be $x$

Alicia travels with the wind is 1 hours 20 minutes

Alicia travels against the wind is 2 hours 30 minutes

Therefore, the total time can be obtained as

1 hours 20 minutes $+$ 2 hours 30 minutes

  $=$ 3 hours 50 minutes

Hence, the total time to take to complete the trip is 3 hours 50 minutes

d

To determine

To write and solve the rational equation to determine the speed of the wind..

Answer to Problem 10CYU

The speed of the wind is 3.5 miles per hours.

Explanation of Solution

Given information :

Alicia’s average speed riding her bikes is 11.5 miles per hours and she take round trip of 40 miles. She takes 1 hours and 20 minutes with the wind and 2 hours 30 minutes against the wind.

Let the speed of the wind be $x$

Alicia’s time against the wind is $\frac{20}{11.5-x}$ .

Alicia’s time with the wind is $\frac{20}{11.5+x}$ .

The total time to take to complete the trip is 3 hours 50 minutes, which is equal to

  $\frac{20}{11.5-x}+\frac{20}{11.5+x}$

Therefore, the equation obtained as $\frac{20}{11.5-x}+\frac{20}{11.5+x}=\frac{23}{6}$

So, the solution of the rational equation to determine the speed of the wind is

  $\begin{array}{l}\frac{20}{11.5-x}+\frac{20}{11.5+x}=\frac{23}{6}\\ \frac{20\left(11.5+x\right)+20\left(11.5-x\right)}{{\left(11.5\right)}^2-{\left(x\right)}^2}=\frac{23}{6}\\ \frac{230+20x+230-20x}{132.25-x^2}=\frac{23}{6}\\ \frac{460}{132.25-x^2}=\frac{23}{6}\\ 2760=3041.75-23x^2\\ 23x^2=3041.75-2760\\ 23x^2=281.75\\ x^2=\frac{281.75}{23}\\ x^2=12.25\\ x=3.5\end{array}$

Hence, the speed of the wind is 3.5 miles per hours.