+1218 A plane flies from Penthaven to Jackson and then back to Penthaven. When there is no wind, the round trip takes 3 hours and 36 minutes, but when there is a wind blowing from Penthaven to Jackson at 30 miles per hour, the trip takes 3 hours and 45 minutes. How many miles is the distance from Penthaven to Jackson?
(Assume that the plane flies at a constant speed, and that the turnaround time is negligible.)
+129847 3 hrs 36 min = 3 + 3/5 hrs = 18/5 hrs
3hrs 45 min = 3 + 3/4 hrs = 15/4 hrs
Let the distance between the towns = D
So....the round trip = 2D
Let the speed of the plane in still air = S
With the wind the the plane travels at S + 30 mph
Against the wind the plane travels at S - 30 mph
So...in still air we have this equation using Distance / Rate = Time
2D / S = 18/5
2D = (18/5)S
S = (10/18)D = (5/9)D (1)
With the wind blowing we have this
D / (S + 30) + D/ (S -30) = 15/4 (2)
Sub (1) into (2)
D / (5/9 D + 30) + D /( 5/9 D - 30) = 15/4 simplify
D(5/9D -30) + D(5/9D + 30) = (15/4) (5/9D + 30) (5/9D -30)
(10/9)D^2 = (15/4) ( 25/81 D - 900)
(10/9)D^2 = (125/108)D^2 -3375
(125/108 - 10/9)D^2 = 3375
(125 / 108 - 120 /108)D^2 = 3375
(5/108)D^2 = 3375
D^2 = 108 / 5 * 3375
D^2 = 72900 take the square root
D = 270 miles
