Chapter 3.6, Problem 13P

To determine

To find:theplane’s air speed and the distance to the nearby town.

Answer to Problem 13P

The plane’s air speed is 60 km per hour and the distance to the nearby town is 105 km.

Explanation of Solution

Given information:

With a head wind, the plane traveled the distance in 2 hours 20 minutes,

With a trial wind, the plane traveled the distance in 1 hours 24 minutes,

The speed of the wind is 15 km/h.

Calculation:

Let x be the air speed of the plane.

With a tail wind, the ground speed is $x+15$ .

With a head wind, the ground speed is $x-15$ .

Note that, the distance traveled by plane is multiplication of ground of speed and time.

The time 1 h 24 m can be changed into hours,

  $\begin{array}{c}\text{Time}=1\text{ h +24 min}\\ \text{=1 h }\text{+}\frac{24}{60}\text{ h}\\ \text{=1+}\frac{2}{5}\\ =\frac{5+2}{5}\\ =\frac{7}{5}\end{array}$

With a trial wind, the plane traveled the distance in 1 hours 24 minutes,

The time 2 h 20 m can be changed into hours,

Here, the distance $\text{D =}\left(x+15\right)\frac{7}{2}$ .

  $\begin{array}{c}\text{Time}=2\text{ h +20 min}\\ \text{=2 h }\text{+}\frac{20}{60}\text{ h}\\ \text{=2+}\frac{1}{3}\\ =\frac{6+1}{3}\\ =\frac{7}{3}\end{array}$

With a head wind, the plane traveled the distance in 2 hours 20 minutes,

Here, the distance $\text{D =}\left(x-15\right)\frac{7}{3}$

Substitute $\text{D =}\left(x-15\right)\frac{7}{3}$ in $\text{D =}\left(x+15\right)\frac{7}{2}$ and find the value of x ,

  $\begin{array}{l}\left(x-15\right)\frac{\cancel{7}}{3}\text{ =}\left(x+15\right)\frac{\cancel{7}}{5}\\ 5\left(x-15\right)=3\left(x+15\right)\\ 5x-75=3x+45\\ 5x-3x=45+75\\ 2x=120\\ x=60\end{array}$

Thus, the plane’s air speed is 60 km per hour.

Substitute $x=60$ in $\text{D =}\left(x-15\right)\frac{7}{3}$ and find the distance,

  $\begin{array}{c}\text{D =}\left(60-15\right)\frac{7}{3}\\ =45\left(\frac{7}{3}\right)\\ =15\left(7\right)\\ =105\end{array}$

Thus, the distance to the nearby town is 105 km.

Therefore, the plane’s air speed is 60 km per hour and the distance to the nearby town is 105 km.