Flying with the wind, a plane traveled 570 miles in 3 hours. Flying against the wind, the plane traveled the same distance in 5 hours. Find the rate of the plane in calm air and the rate of the wind.

Question

Flying with the wind, a plane traveled 570 miles in 3 hours. Flying against the wind, the plane traveled the same distance in 5 hours. Find the rate of the plane in calm air and the rate of the wind.

Rate of plane ______mph
Rate of wind ________mph

Answer 1

let the speed of the plane in still air be x mph

let the speed of the wind be y mph

then 570/(x+y) = 3 --->3x + 3y = 570
x + y = 190

570/(x-y) = 5 ---> 5x - 5y = 570
x - y = 114
add them
2x = 304
x = 152
back in the first:
152 + y = 190
y = 38

Answer 2

To solve this problem, we can use the concept of relative speed.

Let's denote the rate of the plane in calm air as 'p' (in mph), and the rate of the wind as 'w' (in mph).

When the plane is flying with the wind, the effective speed of the plane is the sum of its rate in calm air and the rate of the wind. Therefore, the speed of the plane with the wind is (p + w) mph.

We are given that the plane traveled 570 miles in 3 hours when flying with the wind. Using the formula Distance = Speed × Time, we can write the equation:

570 = (p + w) × 3 ------ (Equation 1)

Similarly, when the plane is flying against the wind, the effective speed of the plane is the difference between its rate in calm air and the rate of the wind. Therefore, the speed of the plane against the wind is (p - w) mph.

We are also given that the plane traveled the same distance of 570 miles in 5 hours when flying against the wind. Using the formula Distance = Speed × Time, we can write the equation:

570 = (p - w) × 5 ------ (Equation 2)

Now we have a system of two equations (Equation 1 and Equation 2) with two variables (p and w). We can solve this system of equations simultaneously to find the values of p and w.

Let's solve the system of equations:

From Equation 1, we can simplify it by dividing both sides by 3:
190 = p + w ------ (Equation 3)

Now let's solve Equations 2 and 3 simultaneously:

Substitute the value of (p + w) from Equation 3 into Equation 2:
570 = (p - w) × 5
570 = (190 - w) × 5
Divide both sides by 5:
114 = 190 - w
Rearrange the equation:
w = 190 - 114
w = 76

Now substitute the value of w = 76 into Equation 3 to find p:
190 = p + 76
p = 190 - 76
p = 114

Therefore, the rate of the plane in calm air is 114 mph, and the rate of the wind is 76 mph.

Rate of plane = 114 mph
Rate of wind = 76 mph