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# Algebra

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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.)

Jun 2, 2024

#1
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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

Jun 3, 2024