A short airplane trip between two cities took 30 minutes when traveling with the wind. The return trip took 45 minutes when traveling against the wind. If the speed of the plane with no wind is 320 mph, find the speed of the wind and the distance between the two cities (hint: pay attention to units).
speed of wind --- x mph
speed of plane going with the wind = 320 + x mph
speed of plane going against the wind = 320-x mph
distance with the wind = (320+x)(1/2)
distance against the wind = (320-x)(3/4)
but those are the same, so
(1/2)(320+x) = (3/4)(320-x)
times 4
2(320+x) = 3(320-x)
640 + 2x = 960 - 3x
5x = 320
x = 64
the speed of the wind is 64 mph
distance = (1/2)(320-x) = 128 miles
Thank you
To find the speed of the wind and the distance between the two cities, we can use the formula:
Speed of the plane with no wind (P) = Speed of the plane in still air (S) + Speed of the wind (W)
Let's assume the speed of the wind is W and the distance between the two cities is D.
When traveling with the wind, the plane's effective speed is increased by the speed of the wind. So, the time taken for this trip is D / (S + W) = 30 minutes.
When traveling against the wind, the plane's effective speed is decreased by the speed of the wind. So, the time taken for this trip is D / (S - W) = 45 minutes.
We can convert the time from minutes to hours by dividing by 60.
Now, let's solve the two equations:
D / (S + W) = 30 / 60 (convert minutes to hours, 30 minutes = 0.5 hours)
D / (S - W) = 45 / 60 (convert minutes to hours, 45 minutes = 0.75 hours)
Since we know the speed of the plane with no wind (320 mph), we can substitute this value into the equations:
D / (320 + W) = 0.5
D / (320 - W) = 0.75
Now, we can solve these equations simultaneously to find the values of W (speed of the wind) and D (distance between the two cities).
Rearranging the first equation, we get:
D = 0.5 * (320 + W)
Rearranging the second equation, we get:
D = 0.75 * (320 - W)
Setting the two expressions of D equal to each other:
0.5 * (320 + W) = 0.75 * (320 - W)
Now, solve for W:
(0.5 * 320) + (0.5 * W) = (0.75 * 320) - (0.75 * W)
160 + 0.5W = 240 - 0.75W
Combine like terms:
0.5W + 0.75W = 240 - 160
1.25W = 80
Divide both sides by 1.25 to solve for W:
W = 80 / 1.25
W = 64 mph
So, the speed of the wind is 64 mph.
Now, substitute this value back into one of the original equations to find the distance D:
D / (320 - 64) = 0.75
D / 256 = 0.75
Multiply both sides by 256:
D = 256 * 0.75
D = 192 miles
Therefore, the distance between the two cities is 192 miles.