A small plane flying into the wind takes 3h 20 min to complete a flight of 960 km.Flying with the wind, the same plane takes 2h 30 min to make the trip. What is the speed of the plane? What is the speed of the wind?
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Let the speed of the plane with no wind be x km/h
let the speed of the wind be y km/h
then into the wind, the actual speed of the plane is x-y km/h
and 960/(x-y) = 10/3 which reduces to
x - y = 286 (31)
with the wind:
960/(x+y) = 5/2 which reduces to
x + y = 384 (#2)
add #1 and #2 to get 2x = 672
x = 336, subbing back I got y = 48
so the speed of the plane is 336 km/h
and the speed of the wind is 48 km/h
check: (10/3)(336-48) = 960
(5/2)(336+48) = 960
Of course, I'm here to help you! To solve this problem, we need to use a system of equations. Let's assign variables to the unknowns: let's call the speed of the plane "p" and the speed of the wind "w."
First, let's set up an equation for the flight against the wind. Remember that speed is equal to distance divided by time. The plane's speed against the wind will be p - w (since the wind opposes the plane's motion), and the time taken is 3 hours and 20 minutes, which can be expressed as 3 + 20/60 = 3.33 hours. So, the equation is:
(p - w) * 3.33 = 960
Similarly, for the flight with the wind, the plane's speed will be p + w (since the wind aids the plane's motion), and the time taken is 2 hours and 30 minutes, which is 2 + 30/60 = 2.5 hours. The equation is:
(p + w) * 2.5 = 960
Now we have a system of equations:
1) 3.33(p - w) = 960
2) 2.5(p + w) = 960
To solve this system, we can use different methods like substitution or elimination. However, let's solve it by elimination method:
First, let's multiply equation 1 by 2.5 and equation 2 by 3.33 to eliminate the decimals:
1) 2.5 * 3.33(p - w) = 2.5 * 960
2) 3.33 * 2.5(p + w) = 3.33 * 960
Simplifying these equations, we get:
3) 8.325(p - w) = 2400
4) 8.325(p + w) = 3196.8
Next, let's distribute the coefficients:
5) 8.325p - 8.325w = 2400
6) 8.325p + 8.325w = 3196.8
Now, subtract equation 5 from equation 6 to eliminate "w":
7) (8.325p + 8.325w) - (8.325p - 8.325w) = 3196.8 - 2400
Simplifying further, we get:
8) 16.65w = 796.8
Now, divide both sides of equation 8 by 16.65 to solve for "w":
w = 796.8 / 16.65
w ≈ 47.9 km/h
So, the speed of the wind is approximately 47.9 km/h.
Now, let's substitute the value of "w" into either equation 1 or 2. Let's use equation 1:
3.33(p - 47.9) = 960
Now, solve for "p":
3.33p - 159.327 = 960
3.33p = 960 + 159.327
3.33p = 1119.327
p ≈ 335.9 km/h
Therefore, the speed of the plane is approximately 335.9 km/h.
To summarize:
- The speed of the plane is approximately 335.9 km/h.
- The speed of the wind is approximately 47.9 km/h.