A plane flies 540 miles for 2 hours in the direction of the wind. Then it turns around and flies 690 miles
for 3 hours against the same wind. Find the speed of the plane in still air and the wind speed.
speed of plane in still air --- x mph
speed of wind ------ y mph
rate into the wind = x-y
rate with the wind = x+y
2(x+y) = 540
x + y = 270
3(x-y) = 690
x-y = 230
add them:
2x = 500
x = 250
then y = 20
Well, if the plane flies in the direction of the wind, it's like throwing a clown on a trampoline – it will bounce back a bit slower! So, let's call the speed of the plane in still air "P" and the speed of the wind "W".
When the plane flies with the wind, its effective speed is the sum of the plane's speed in still air and the speed of the wind, so we have P + W.
When the plane flies against the wind, its effective speed is the difference between the plane's speed in still air and the speed of the wind, so we have P - W.
We're given the distances and times, so we can set up two equations to solve for P and W.
First, the plane flies 540 miles in 2 hours in the direction of the wind, so:
540 = (P + W) * 2
And second, the plane flies 690 miles in 3 hours against the wind, so:
690 = (P - W) * 3
Now let's solve this clownish puzzle! Solving the first equation for P + W gives us:
P + W = 270
And solving the second equation for P - W gives us:
P - W = 230
Now, we have two equations with two variables. Let's add these equations together:
(P + W) + (P - W) = 270 + 230
This simplifies to:
2P = 500
Now, divide both sides by 2:
P = 250
Hooray! We have found the speed of the plane in still air, which is 250 miles per hour.
Now, we can substitute this value back into either of the original equations to find the wind speed. Let's use the first equation:
540 = (250 + W) * 2
Divide both sides by 2:
270 = 250 + W
Subtract 250 from both sides:
W = 20
So, the wind speed is 20 miles per hour.
To summarize, the speed of the plane in still air is 250 miles per hour, and the wind speed is 20 miles per hour.
To solve this problem, we can use the formula for the speed of an object where its speed is equal to the distance traveled divided by the time taken.
Let's denote the speed of the plane in still air as "p" and the speed of the wind as "w."
Step 1: Find the speed of the plane with the wind
The speed of the plane with the wind can be calculated as:
p + w
Given that the plane flies 540 miles for 2 hours with the wind, we can find the formula:
540 miles = (p + w) × 2 hours
Step 2: Find the speed of the plane against the wind
The speed of the plane against the wind can be calculated as:
p - w
Given that the plane flies 690 miles for 3 hours against the wind, we can find the formula:
690 miles = (p - w) × 3 hours
Step 3: Solve the equations
Now, we have a system of equations to solve simultaneously:
540 = (p + w) × 2 -- Equation 1
690 = (p - w) × 3 -- Equation 2
Step 4: Eliminate variables
Let's simplify Equation 1 by dividing both sides by 2:
270 = p + w -- Equation 1a
Now, simplify Equation 2 by dividing both sides by 3:
230 = p - w -- Equation 2a
Step 5: Solve for p (speed of the plane in still air)
Add Equation 1a and Equation 2a to eliminate "w":
270 + 230 = (p + w) + (p - w)
500 = 2p
Divide both sides by 2:
250 = p
Step 6: Solve for w (wind speed)
Substitute the value of p back into Equation 1a or Equation 2a:
230 = 250 - w
Rearrange the equation to solve for w:
w = 250 - 230
w = 20
Therefore, the speed of the plane in still air is 250 mph, and the wind speed is 20 mph.
To find the speed of the plane in still air and the wind speed, we can set up a system of equations. Let's denote the speed of the plane in still air as "p" and the speed of the wind as "w".
When flying with the wind, the effective speed of the plane is enhanced by the speed of the wind. So, the equation for the distance flown with the wind is:
Distance = Rate * Time
540 = (p + w) * 2
When flying against the wind, the effective speed of the plane is reduced by the speed of the wind. So, the equation for the distance flown against the wind is:
Distance = Rate * Time
690 = (p - w) * 3
Now we have a system of equations:
1. 540 = (p + w) * 2
2. 690 = (p - w) * 3
We can solve this system of equations to find the values of p and w.
First, let's simplify equation 1:
540 = 2p + 2w (divided both sides of the equation by 2)
Next, let's simplify equation 2:
690 = 3p - 3w (divided both sides of the equation by 3)
Now we have the following system of simplified equations:
3. 540 = 2p + 2w
4. 690 = 3p - 3w
To eliminate w, we can multiply equation 3 by 3 and equation 4 by 2:
5. 1620 = 6p + 6w
6. 1380 = 6p - 6w
Adding equations 5 and 6 eliminates w:
7. 3000 = 12p
8. p = 3000 / 12
p = 250
Now we can substitute the value of p into one of the original equations to find the value of w. Let's use equation 3:
540 = (p + w) * 2
540 = (250 + w) * 2
540 = 500 + 2w
540 - 500 = 2w
40 = 2w
w = 40 / 2
w = 20
Therefore, the speed of the plane in still air is 250 mph and the speed of the wind is 20 mph.