Let us divide this problem into two parts:
1) Sam rides up the mountain.
2) Sam rides down the mountain.
1.
Since speed is distance over time, as:
[tex]v = \frac{s}{t} [/tex]
Therefore, distance would be:
[tex]s_{up} = v_{up} * t_{up} [/tex]
Where s = distance,
v = speed,
t = time.
In the problem, Sam's speed while riding up is v = 6 miles/hour = (6 * 1609.34 / 60) = 160.934 meters/second(in SI Units). Plug this value in the above equation, you would get:
[tex]s_{up} = 160.934 * t_{up}[/tex] --- (A)
2.
As Sam rides down the mountain, the speed given is:
[tex]v_{down} = 54 miles/h[/tex]
Convert it in SI units; the speed would be in SI unit:
v = 54 miles/hour = (54 * 1609.34 / 60) = 1448.406 meters/second(in SI Units). Plug this value in the distance equation, you would get:
[tex]s_{down} = 1448.406 * t_{down}[/tex]
Since the [tex]s_{up} = s_{down}[/tex], therefore,
[tex]160.934 * t_{up} = 1448.406 * t_{down}[/tex]
=> [tex] t_{up} = 9 t_{down}[/tex]
Now the condition is that the whole trip, up and down, takes 40 minutes(2400seconds), it means:
[tex]t_{up} + t_{down} = 2400[/tex]
Plug in the value of [tex]t_{up} [/tex] in the above equation, you would get:
[tex]t_{down} = 240[/tex]
Therefore,
[tex]s_{down} = 1448.406*240[/tex]
[tex]s_{down} = 347617.44[/tex] meters (in relation to seconds)
[tex]s_{down} = 5793.624[/tex] meters (in relation to hours)
Now the last step is to convert meters into miles, you would get:
[tex]s_{down} = 5793.624/1609.34 = 3.6miles[/tex]
So the answer is 3.6miles.
