# Against the wind a small plane flew 210 miles in 1 hour and 10 minutes. The return trip took only 50 minutes. How do you determine the speed of the plane?

Oct 3, 2015

Speed of the plane $= 3.6$ miles/minute $= 216$ miles/hour

#### Explanation:

If ${s}_{p}$ is the speed of the plane (in miles per minute)
and ${s}_{w}$ is the speed of the wind (in miles per minute)

We are told
$\textcolor{w h i t e}{\text{XXX}} {s}_{p} - {s}_{w} = \frac{210}{70}$
$\textcolor{w h i t e}{\text{XXX}} {s}_{p} + {s}_{w} = \frac{210}{50}$

$\textcolor{w h i t e}{\text{XXX}} 2 {s}_{p} = \frac{21 \cancel{0}}{7 \cancel{0}} + \frac{21 \cancel{0}}{5 \cancel{0}}$

$\textcolor{w h i t e}{\text{XXX}} 2 {s}_{p} = \frac{105 + 147}{35}$

$\textcolor{w h i t e}{\text{XXX}} 2 {s}_{p} = \frac{252}{35}$

$\textcolor{w h i t e}{\text{XXX}} {s}_{p} = \frac{126}{35} = \frac{18}{5} = 3.6$

$3.6$ miles/ minute
$\textcolor{w h i t e}{\text{XXX")= 3.6 ("miles")/(cancel("minute"))*60 (cancel("minutes"))/("hour") = 216 ("miles")/("hour}}$

Oct 3, 2015

I found $3.6 \frac{\text{miles}}{\min}$

#### Explanation:

Call ${v}_{p}$ the plane velocity and ${v}_{w}$ the wind velocity.
Against the wind you have:
${v}_{p} - {v}_{w} = \frac{210}{70}$
where I used the fact that velocity=distance/time and changed time in minutes.
When the wind is in the same direction you get:
${v}_{p} + {v}_{w} = \frac{210}{50}$

From the first equation:
${v}_{p} = {v}_{w} + \frac{210}{70} = {v}_{w} + 3$ substitute into the second:
${v}_{w} + 3 + {v}_{w} = 4.2$
$2 {v}_{w} = 1.2$
${v}_{w} = \frac{1.2}{2} = 0.6 \frac{\text{miles}}{\min}$
and using this into: ${v}_{p} = {v}_{w} + 3$
${v}_{p} = 3.6 \frac{\text{miles}}{\min}$