An aircraft flew 3hrs with the wind. The return trip took 4hrs against the wind. If the speed of the plane in still air is 180mph more than the speed of the wind, find the wind speed and the speed of the plane in still air.

distance=rate*time

d=rt

wind speed=w
plane spee=180+w

d=[(180+w)+w]*3 With wind
d=[(180+w)-w]*4 Against wind
[(180+w)+w]*3=[(180+w)-w)*4

(180+2w)3=180*4
540+6w=720
6w=180
w=30

The wind speed is 30 mph.

The plane in still air is 180+30, or 210.

Well, I guess the plane must have gotten tired on its return trip. It's always harder to go against the wind, right?

Let's say the speed of the plane in still air is P mph, and the speed of the wind is W mph.

When the plane flies with the wind, its effective speed is increased by the wind speed. So, the speed with the wind becomes P + W mph.

On the return trip, when the plane flies against the wind, its effective speed is reduced by the wind speed. So, the speed against the wind becomes P - W mph.

Given that the plane flew 3 hours with the wind at P + W mph, and 4 hours against the wind at P - W mph, we can form the following equations:

Distance = Speed × Time

For the trip with the wind: Distance = (P + W) × 3

For the return trip against the wind: Distance = (P - W) × 4

Since the distances are the same (assuming the plane didn't take any side trips!), we can set up an equation:

(P + W) × 3 = (P - W) × 4

Now, we can solve this equation to find the values of P and W.

Expanding and simplifying the equation, we get:

3P + 3W = 4P - 4W

Moving similar terms to one side, we get:

4W + 3W = 4P - 3P

7W = P

Now, we know that the speed of the plane in still air is 180 mph more than the speed of the wind. So, P = W + 180.

Substituting this into the equation above, we get:

7W = W + 180

Subtracting W from both sides:

6W = 180

Dividing both sides by 6:

W = 30

Now that we know the wind speed is 30 mph, we can substitute this back into the equation P = W + 180 to find the speed of the plane in still air:

P = 30 + 180

P = 210

So, the speed of the plane in still air is 210 mph, and the wind speed is 30 mph.

Let's assume the speed of the plane in still air is represented by "p" and the speed of the wind is represented by "w."

When flying with the wind, the plane's effective speed is increased by the speed of the wind. Therefore, the speed of the plane flying with the wind can be represented as "p + w."

When flying against the wind, the plane's effective speed is decreased by the speed of the wind. Therefore, the speed of the plane flying against the wind can be represented as "p - w."

Given that the aircraft flew 3 hours with the wind and 4 hours against the wind, we can use the formula: distance = speed × time to find the respective distances:

Distance flown with the wind = (p + w) × 3
Distance flown against the wind = (p - w) × 4

Since the distances are equal, we can set up an equation:

(p + w) × 3 = (p - w) × 4

Expanding this equation, we get:

3p + 3w = 4p - 4w

Rearranging the terms, we have:

3w + 4w = 4p - 3p
7w = p

We also know that the speed of the plane in still air is 180 mph more than the speed of the wind:

p = w + 180

Substituting this into the previous equation, we have:

7w = w + 180

Simplifying the equation:

7w - w = 180
6w = 180

Dividing both sides by 6:

w = 30

So, the wind speed is 30 mph.

To find the speed of the plane in still air, we use the equation we derived earlier:

p = 7w
p = 7(30)
p = 210

Therefore, the speed of the plane in still air is 210 mph.

To solve this problem, we can set up a system of equations using the given information.

Let's denote the speed of the plane in still air as P and the speed of the wind as W.

Distance = Speed × Time

For the first leg of the trip with the wind, the distance traveled is equal to the speed of the plane in still air plus the speed of the wind multiplied by the time:

Distance with the wind = (P + W) × 3

For the return trip against the wind, the distance traveled is equal to the speed of the plane in still air minus the speed of the wind multiplied by the time:

Distance against the wind = (P - W) × 4

Since the distance traveled in both directions is the same (as it's a round trip), we can set up the following equation:

(P + W) × 3 = (P - W) × 4

Now, let's solve this equation to find the values of P (plane speed in still air) and W (wind speed):

3P + 3W = 4P - 4W
3W + 4W = 4P - 3P
7W = P

Given that the speed of the plane in still air is 180 mph more than the speed of the wind, we can rewrite P as:

P = W + 180

Substituting this value into the equation we found earlier:

7W = W + 180

Simplifying the equation by subtracting W from both sides:

6W = 180

Dividing both sides of the equation by 6:

W = 30

Now that we have the value of W (wind speed), we can substitute it back into the equation P = W + 180:

P = 30 + 180
P = 210

Therefore, the wind speed is 30 mph and the speed of the plane in still air is 210 mph.