A plane is traveling at a speed of 160 mph in still air. Flying with a tail wind, the plane is clocked over a distance of 800 miles. Flying against a headwind it takes 3 more hours to complete the return trip. What was the wind velocity?
To find the wind velocity, we need to set up a system of equations based on the given information.
Let's assume that the speed of the wind is 'w' mph.
When the plane is flying with the tailwind, the effective speed of the plane will be its normal speed plus the speed of the wind. Therefore, the effective speed will be:
effective speed = 160 mph + w mph
The time it takes to fly a distance of 800 miles at this speed can be calculated using the formula:
time = distance / speed
So, the time taken with the tailwind is:
800 miles / (160 mph + w mph)
Now, when the plane is flying against the headwind, the effective speed will be the normal speed of the plane minus the speed of the wind. Therefore, the effective speed will be:
effective speed = 160 mph - w mph
The time it takes to fly a distance of 800 miles at this speed is:
800 miles / (160 mph - w mph)
According to the problem, it takes 3 more hours to complete the return trip (flying against the headwind) compared to flying with the tailwind. So we set up the equation:
800 miles / (160 mph + w mph) + 3 hours = 800 miles / (160 mph - w mph)
Now we can solve this equation to find the value of 'w', which represents the wind velocity.
First, let's simplify the equation by multiplying both sides of the equation by (160 mph + w mph) and (160 mph - w mph) to eliminate the denominators:
800(160 mph - w mph) + 3(160 mph + w mph) = 800(160 mph + w mph)
Next, let's expand and simplify the equation:
128000 mph - 800w mph + 480 mph + 3w mph = 128000 mph + 800w mph
Combine like terms:
128480 mph - 797w mph = 128000 mph + 800w mph
Now, let's isolate the variable:
-797w mph - 800w mph = 128000 mph - 128480 mph
Combine like terms:
-1597w mph = -480 mph
Divide both sides by -1597 mph to solve for 'w':
w = (-480 mph) / (-1597 mph)
Simplifying the division:
w ≈ 0.3004 mph
Therefore, the wind velocity is approximately 0.3004 mph.
wind speed ---- x mph
speed going with the wind = 160+x
speed going against the wind = 160-x
800/(160-x) - 800(160+x) = 3
times (160-x)(160+x)
800(160+x) - 800(160-x) = 3(160+x)(160-x)
800x + 800x = 76800 - 3x^2
3x^2 + 1600x - 76800 = 0
x = (-1600 ± √3481600)/6
x = appr 44.3 mph or some negative x
The speed of the wind is appr 44.3 mph
check:
800/115.7 - 800/204.3
= 2.9986 , not bad