An airplane flies 3024 miles into the wind for 14 hours. The return trip takes 9 hours. Find the speed of the airplane without a wind and the speed of the wind.
If the plane's speed is p and the wind speed is w, then we have
14(p-w) = 3024
9(p+w) = 3024
To solve this problem, we can use the concept of relative speed. Let's denote the speed of the airplane without a wind as "x" and the speed of the wind as "w."
When an airplane is flying into the wind, its effective speed decreases. When the airplane is flying with the wind, its effective speed increases. Therefore, we need to consider the relative speed of the airplane with respect to the wind for both legs of the trip.
Given that the airplane flew 3024 miles into the wind for 14 hours, we can set up the equation:
Distance = Speed x Time
3024 = (x - w) x 14
Similarly, for the return trip of 9 hours, we can set up another equation:
Distance = Speed x Time
3024 = (x + w) x 9
Now, we have a system of two equations that we can solve to find the values of x (speed without wind) and w (speed of the wind).
Let's start by expanding the equations:
14x - 14w = 3024 ---- (Equation 1)
9x + 9w = 3024 ---- (Equation 2)
Now, we can solve this system of equations using any method, such as substitution or elimination.
Let's use the elimination method to solve this system:
We can multiply Equation 1 by 9 and Equation 2 by 14 to make the coefficients of "w" equal:
126x - 126w = 14 * 3024
126x + 126w = 9 * 3024
Adding the equations, we have:
252x = 14 * 3024 + 9 * 3024
252x = 23 * 3024
Dividing both sides by 252, we get:
x = (23 * 3024) / 252
Now, we can calculate the value of x.
x = (23 * 3024) / 252
x = 27672 / 252
x = 110
Therefore, the speed of the airplane without any wind is 110 miles per hour.
Now, let's find the speed of the wind, denoted as "w."
We can substitute the value of x = 110 into either Equation 1 or Equation 2 to find w.
Let's use Equation 1:
14x - 14w = 3024
Substituting the value of x as 110:
14 * 110 - 14w = 3024
1540 - 14w = 3024
-14w = 3024 - 1540
-14w = 1484
Dividing both sides by -14, we get:
w = 1484 / -14
w ≈ -106
Therefore, the speed of the wind is approximately 106 miles per hour.
To summarize:
- The speed of the airplane without a wind is 110 miles per hour.
- The speed of the wind is approximately 106 miles per hour.