An airplane took 3 hours to fly 600 miles against a headwind. The return trip with the wind took 2 hours. Find the speed of the plane in still air and the speed of the wind?
Speed of the plane: 250 mph
Speed of the wind: 50 mph
Let p = the speed of the plane and w = the speed of the wind
It takes the plane 3 hours to go 600 miles when against the headwind and 2 hours to go 600 miles with the headwind. So we set up a system of equations.
Solving for the left sides we get:
200mph = p  w
300mph = p + w
Now solve for one variable in either equation. I'll solve for x in the first equation:
200mph = p  w
Add w to both sides: p = 200mph + w
Now we can substitute the x that we found in the first equation into the second equation so we can solve for w:
300mph = (200mph + w) + w
Combine like terms: 300mph = 200mph + 2w
Subtract 200mph on both sides: 100mph = 2w
Divide by 2: 50mph = w
So the speed of the wind is 50mph. Now plug the value we just found back in to either equation to find the speed of the plane, I'll plug it into the first equation:
200mph = p  50mph
Add 50mph on both sides: 250mph = p
So the speed of the plane in still air is 250mph.
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Ahhh, yes. Relative wind problems. There was a time when I had to do these in my head, while flying the plane.
Anyway, you have 2 equations in 2 unknowns. Let p be the plane's velocity, and w be the wind speed.
And remember, d=rt, where d = distance, r = rate, and t = time.
or:
Let's take the second equation, and write w as a function of p.
...now substitute this back in the first equation:
now, go back to the derivation we had for w:
CHECK YOUR WORK:
Upwind, your speed is 200.
GOOD LUCK.
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Let ( p ) be the speed of the plane in still air and ( w ) be the speed of the wind.

Against the wind: [ p  w = \frac{600}{3} ]

With the wind: [ p + w = \frac{600}{2} ]
Solve the system of equations to find the values of ( p ) and ( w ).
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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