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# PLANE PROB, RATES

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8
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+299

A plane flies from Penthaven to Jackson and then back to Penthaven. When there is no wind, the round trip takes 5 hours and 20 minutes, but when there is a wind blowing from Penthaven to Jackson at 75 miles per hour, the trip takes 6 hours. How many miles is the distance from Penthaven to Jackson?

Mar 2, 2024

#1
+129743
+1

5 hrs 20 min =  5 + 1/3 hrs =  (16/3)hrs

Call the rate of the plane in still air =  R

With the wind the rate is R + 75

Against the wnid the rate is R - 75

Call the distance between the cities D

D/R = T

In the first instance

Total Distance / Rate in still air = 16/3 hrs

2D / R   =  (16/3)

D/R = 8/3

D = (8/3)R      (1)

In the second instance

Time from Penthaven to Jackson  + Time from Jackson to Penthaven  =  6  hrs

D / ( R + 75 )  +  D / ( R - 75)  =  6      (2)

Sub (1)  into (2)

(8/3)R / ( R + 75)  + (8/3)R / ( R -75)   = 6

(8/3)R ( R -75)  + (8/3)R (R + 75)  =  6 ( R+ 75)(R -75)

(8/3)R^2 - 200R + (8/3)R^2 + 200R  =  6 ( R^2 - 5625)

(16/3)R^2  = 6R^2 - 33750

R^2 ( 6 -16/3)  = 33750

(2/3)R^2 = 33750

R^2  = 50625

R=  225

The distance between the  cities

D = (8/3)225 =   600  miles

Mar 2, 2024
edited by CPhill  Mar 3, 2024
#2
+299
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The problem says round trip, from Penthaven to Jackson and back. Then the problem says that the wind is blowing from Penthaven to Jackson during our round trip.

Lilliam0216  Mar 2, 2024
#3
+129743
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Sorry....I thought this problem was similar to another one asked on here earlier.....see my corrected answer ....

CPhill  Mar 3, 2024