Answer: approximately 43.57337308309 mph
Round this value however you need to
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Step-by-step explanation:
w = speed of the wind
The plane travels at a speed of 180 mph in still air. With a tailwind, the plane speeds up (the wind is coming from the tail to push the plane forward more), so the plane is now traveling at a speed of 180+w miles per hour. It travels 700 miles
d = r*t
700 = (180+w)*t
t = 700/(180+w)
This is the time it takes when the wind speeds up the plane.
In contrast, the headwind slows the plane down because now the wind is going against the plane. Headwinds attack the head of the plane and push against the plane. With a headwind, the plane is now going 180-w miles per hour. It travels for 2 hours longer to do the return trip (of 700 miles), so,
d = r*t
700 = (180-w)*(t+2)
700 = 180t + 360 - w*t - 2w
700 = 180( 700/(180+w) ) + 360 - w*( 700/(180+w) ) - 2w
700 = 126000/(180+w) + 360 - 700w/(180+w) - 2w
700(180+w) = 126000 + 360(180+w) - 700w - 2w(180+w)
126000+700w = 126000+64800+360w-700w-360w-2w^2
126000+700w = -2w^2 - 700w + 190800
-2w^2 - 700w + 190800 = 126000+700w
-2w^2 - 700w + 190800 - 126000-700w = 0
-2w^2 - 1400w + 64800 = 0
Use a graphing calculator or the quadratic formula to find the approximate solutions are
w = -743.5733730831, w = 43.57337308309
Ignore the negative value as it makes no sense to have a negative wind speed.
The only practical answer is approximately 43.57337308309 mph