Question 164682: Dave and Sandy fly from Philly to Chicago, it takes 2 hours to go west and 1 and a half hours to fly east. The trip is 780 one way. If the wind speed is the same on each trip, find the speed of the wind and find the speed of the plane in the still air.
Answer by gonzo(654)   (Show Source):
You can put this solution on YOUR website! D = 780 miles
let A = airspeed
let W = windspeed
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since it takes more time to go west than east, it's fair to say that the wind was against the airplane going west and with the airplane going east.
TW = time going west = 2 hours
TE = time going east = 1.5 hours
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since the rate of travel times the duration of travel equals the distance, the equations for going west and going east are:
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TW * (A-W) = 780
TE * (A+W) = 780
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substituting for TW and TE makes the equations become:
2 * (A-W) = 780 (original first equation)
1.5 * (A+W) = 780 (original second equation)
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since both equations equal 780, then they are equal to each other (transitive property of algebraic operations).
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2 * (A-W) = 1.5 * (A+W)
removing parentheses:
2*A - 2*W = 1.5*A + 1.5*W
subtracting 1.5*A from both sides of equation and adding 2*W to both sides of equation:
2*A - 1.5*A = 1.5*W + 2*W
combining like terms:
.5*A = 3.5*W
multiplying both sides of equation by 2:
A = 7*W
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original first equation is:
2 * (A-W) = 780
removing parentheses:
2*A - 2*W = 780
substituting 7*W for A:
2*7*W - 2*W = 780
simplifying and combining like terms:
12*W = 780
dividing both sides by 12:
W = 65
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original second equation is:
1.5 * (A+W) = 780
removing parentheses:
1.5*A + 1.5*W = 780
substituting 65 for W:
1.5*A + 1.5*65 = 780
simplifying:
1.5*A + 97.5 = 780
subtracting 97.5 from both sides of equation:
1.5*A = 780 - 97.5
simplifying:
1.5*A = 682.5
dividing both sides of equation by 1.5
A = 682.5/1.5
simplifying:
A = 455
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original first and second equations are:
2 * (A-W) = 780 (original first equation)
1.5 * (A+W) = 780 (original second equation)
substituting 455 for A and 65 for W in original first equation:
2 * (455-65) = 780
simplifying:
2*390 = 780
780 = 780
first equation is good
substituting 455 for A and 65 for W in original second equation:
1.5 * (455+65) = 780
simplifying:
1.5 * (520) = 780
780 = 780
second equation is good.
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answer is:
Airplane speed is 455 miles per hour.
Wind speed is 65 miles per hour.
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