Flying with the wind, a small plane flew 332 mi in 2 h. Flying against the wind, the plane could fly only 318 mi in the same amount of time. Find the rate of the plane in calm air and the rate of the wind
332/2 = 166 = v+c
318/2 = 159 = v-c
2 v = 166+159 = 325 etc
325
To solve this problem, let's assume the rate of the plane in calm air (without the effect of wind) is represented by "p" and the rate of the wind is represented by "w".
When the plane flies with the wind, its effective speed increases due to the wind pushing it forward. Therefore, the plane's speed is p + w (plane's speed + wind's speed).
When the plane flies against the wind, its effective speed decreases due to the wind pushing against it. Therefore, the plane's speed is p - w (plane's speed - wind's speed).
Given that the plane flew 332 miles in 2 hours with the wind, we can set up the equation:
(p + w) * 2 = 332
Simplifying, we get:
2p + 2w = 332 (equation 1)
Similarly, given that the plane could only fly 318 miles in the same 2 hours against the wind, we can set up another equation:
(p - w) * 2 = 318
Simplifying, we get:
2p - 2w = 318 (equation 2)
Now we have a system of equations (equation 1 and equation 2) that we can solve simultaneously to find the values of p and w.
To solve the system of equations, we can either use substitution or elimination method. Let's use the elimination method in this case.
Multiplying equation 1 by 1 and equation 2 by 3, we get:
2p + 2w = 332 (equation 1)
6p - 6w = 954 (equation 2)
Now, add equations 1 and 2:
2p + 2w + 6p - 6w = 332 + 954
This simplifies to:
8p = 1286
Dividing both sides by 8, we get:
p = 160.75
So, the rate of the plane in calm air is 160.75 mph.
To find the rate of the wind, we can substitute the value of p back into equation 1 (or equation 2). Let's use equation 1:
2p + 2w = 332
Substituting the value of p, we get:
2(160.75) + 2w = 332
Simplifying, we get:
321.5 + 2w = 332
Subtracting 321.5 from both sides, we get:
2w = 332 - 321.5
This simplifies to:
2w = 10.5
Dividing both sides by 2, we get:
w = 5.25
So, the rate of the wind is 5.25 mph.
Therefore, the rate of the plane in calm air is 160.75 mph and the rate of the wind is 5.25 mph.