Rod is a pilot for Crossland Airways. He computes his flight time against a headwind for a trip of 2900 miles at 5 hr. The flight would take 4 hr and 50 min if the headwind were half as great. Find the headwind and the plane’s air speed.
Please help me solve. I'm unable to solve. I have no clue!
speed first trip = s - h
speed second trip = s - 0.5 h
distance = rate * time
2900 = (s-h)(5)
2900 = (s-0.5h)(4 50/60) = (s - 0.5h)(4.833)
P = plane's speed.
W = wind speed.
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The plane traveled 2900 miles in 300 minutes (5 hr x 60 min/hr = 300 min).
distance = rate x time
rate = d/t = 2900 miles/300 min = 9.667 mi/min.
One equation is P-W = 9.667 miles/min.
If the wind speed is 1/2 that, it takes 4 hrs and 50 min (290 minutes).
rate = d/t = 2900/290 = 10 miles/min.
Second eqution is P-0.5W = 10.
Two equations in P and W. Solve for P and W. If I didn't make an error (check me out on the equations and arithmetic), P = 10.337 = 10.0 miles/min.
Check.
First flight
10.333 - 0.667 = 9.667 mi/min.
and 9.667 mi/min x 300 min = 2900 miles.
If wind is 1/2 that, it will be 0.337
10.337 - 0.337 = 10 mi/min x 290 min = 2900 miles.
You many change miles/min to miles/hr if you wish but I thought 4 hours and 50 minutes would be better in minutes.
W = 0.667 miles/min.
Two equations in P and W. Solve for P and W. If I didn't make an error (check me out on the equations and arithmetic), P = 10.337 = 10.0 miles/min
I made a typo here. P = 10.337 miles/min.
To solve this problem, we need to set up a system of equations. Let's start by defining the variables:
Let's call the plane's airspeed (which is the speed of the plane relative to the air) "p."
Let's call the headwind speed "h."
Given that the flight time against the headwind for a trip of 2900 miles is 5 hours, we can set up the equation:
2900/(p - h) = 5
We are also given that the flight time would take 4 hours and 50 minutes (or 4.83 hours) if the headwind were half as great. This gives us another equation:
2900/(p - h/2) = 4.83
Now, we have a system of equations. We can solve it by using substitution or elimination method. Let's use the elimination method in this case:
Step 1: Multiply the second equation by 2 to eliminate the fractions:
5800/(p - h) = 9.66 (approximately)
Step 2: Multiply the second equation numerator by (p - h) and multiply the first equation numerator by (p - h/2):
5800 = 9.66(p - h)
2900 = 5(p - h/2)
Step 3: Simplify both equations:
5800 = 9.66p - 9.66h
2900 = 5p - 2.5h
Step 4: Rearrange the equations:
9.66p - 9.66h = 5800
5p - 2.5h = 2900
Step 5: Multiply the second equation by 3.864 (the approximate inverse of 9.66) to eliminate p:
3.864 * (5p - 2.5h) = 3.864 * 2900
Step 6: Simplify the equations:
9.66p - 9.66h = 5800
19.32p - 9.66h = 11203.2
Step 7: Subtract the first equation from the second equation:
19.32p - 9.66h - 9.66p + 9.66h = 11203.2 - 5800
Step 8: Simplify and combine like terms:
9.66p = 5403.2
Step 9: Divide both sides of the equation by 9.66:
p = 558.69 (approximately)
So, the plane's airspeed (or speed relative to the air) is approximately 558.69 mph.
Now, we can substitute this value back into any of the original equations to find the headwind speed. Let's use the first equation:
2900/(558.69 - h) = 5
Step 10: Solve for h:
2900 = 5(558.69 - h)
2900 = 2793.45 - 5h
5h = 2900 - 2793.45
5h = 106.55
h ≈ 21.31
So, the headwind speed is approximately 21.31 mph.
Therefore, the plane's airspeed is approximately 558.69 mph, and the headwind speed is approximately 21.31 mph.