Given, Flying at a constant speed, a plane can travel 800 miles with the wind in the same amount of time it can fly 650 miles against the wind. Find the speed of the plane if the wind blows at 30 mph?
The time is calculated by dividing the distance by the speed.
t = 800 mi / (s + 30) mph
t = 650 mi / (s - 30) mph
The problem states "in the same amount of time," so t is the same in both equations.
800 / (s + 30) = 650 / (s - 30)
We multiply both sides by (s + 30) * (s - 30). That eliminates both denominators.
((s + 30) * (s - 30)) * (800 / (s + 30)) = ((s + 30) * (s - 30)) * (650 / (s - 30))
On the left side, the (s + 30) cancels out; on the right side, the (s - 30) cancels out.
800 * (s - 30) = 650 * (s + 30)
800s - 24000 = 650s + 19500
Subtract 650s from both sides; add 24000 to both sides.
800s - 650s - 24000 + 24000 = 650s - 650s + 19500 + 24000
150s = 43500
Divide both sides by 150.
150s / 150 = 43500 / 150
s = 290
The plane is flying at 290 mph (airspeed)
Flying with the wind, the plane's ground speed is 290 + 30 = 320 mph
800 mi / 320 mph = 2.5 hr
Flying against the wind, the plane's ground speed is 290 - 30 - 260 mph
650 mi / 260 mph = 2.5 hr