Christie pilots her plane for 320 mi against the headwind in 2 hrs . The flight would take 1 hr and 36 minutes with a tailwind of the same speed. Find the headwind and the speed of the plane in still air.
1 hr 36 min = 1.6 hr
p - w = 320 mi / 2 hr = 160 mph
p + w = 320 mi / 1.6 hr = 200 mph
adding equations (to eliminate w) ... 2 p = 36 mph
solve for p , then substitute back to find w
if the wind speed is w and the plane's speed is s, then since distance = speed*time,
2(s-w) = 320
1.6(s+w) = 320
now just solve for s and w
dropped a zero
2 p = 360
To find the headwind and the speed of the plane in still air, we can set up a system of equations based on the given information.
Let's denote the speed of the plane in still air as "P" and the speed of the headwind (or tailwind) as "W".
First, let's calculate the speed of the plane relative to the headwind when flying against it. The speed of the plane relative to the headwind is given by:
P - W
We know that when traveling against the headwind for 320 miles, the time taken is 2 hours (or 2 hours and 0 minutes). Therefore, the speed of the plane relative to the headwind multiplied by the time taken should give us the distance traveled:
(P - W) * 2 = 320
Next, let's calculate the speed of the plane relative to the tailwind when flying with it. The speed of the plane relative to the tailwind is given by:
P + W
We know that when traveling with the tailwind for the same 320 miles, the time taken is 1 hour and 36 minutes. We need to convert the time into hours by dividing the minutes by 60:
1 hour + (36 minutes / 60 minutes per hour) = 1.6 hours
Therefore, the speed of the plane relative to the tailwind multiplied by the time taken should give us the distance traveled:
(P + W) * 1.6 = 320
Now we have a system of two equations:
1) (P - W) * 2 = 320
2) (P + W) * 1.6 = 320
We can solve this system of equations to find the values of P and W.
From equation 1, we can simplify by dividing both sides by 2:
(P - W) = 160
From equation 2, we can simplify by dividing both sides by 1.6:
(P + W) = 200
Now we have a simplified system of equations:
3) P - W = 160
4) P + W = 200
Adding equations 3 and 4 eliminates the variable W:
2P = 360
Dividing both sides by 2:
P = 180
So the speed of the plane in still air is 180 miles per hour.
Substituting this value into equation 4, we can find W:
180 + W = 200
Subtracting 180 from both sides:
W = 200 - 180
W = 20
Therefore, the headwind is 20 miles per hour.