Question 498444: a plane is flying the 2553-mi trip from los angeles to honolulu into a 60-mph headwind. if the speed of the plane in still air is 310 mph, how far from los angeles is the plane's point of no return?
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! Let's first look at the true ground speed of the airplane in both directions.
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When going to Hawaii from Los Angeles the plane is being slowed down by the head wind. In still air, the plane would fly at 310 mph, but the 60 mph head wind will slow that rate down to 250 mph (310 minus 60 = 310).
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However, presuming that somewhere during the flight the plane reverses direction and flies back toward Los Angeles, the true ground speed of the plane will be boosted by the wind pushing it. Therefore, instead of flying at 310 mph, the ground speed will be 370 mph (310 plus 60 = 370).
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We are going to use the formula Distance = Speed * Time or D = S*T for short. But we want to rearrange it to solve for Time. Divide both sides by S to get:
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T = D/S
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Here's what we are going to do. We want to find a point along the route where the time to continue the flight to Hawaii equals the time that it would take to fly back to Los Angeles.
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Let's assume that we fly out from Los Angeles some distance X. Then how far would we still have to fly to get to Hawaii. Since the total distance of the flight is 2553 miles, after flying for X miles we still have 2553 minus X miles to go. How long would it take to finish the flight to Hawaii after we have gone X miles already?
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T = D/S = (2553 - X)/S
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But remember that the speed on the way to Hawaii is 250 mph. Substitute this into the equation and it becomes:
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T = (2553 - X)/250
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That is the equation for the amount of time it would take to finish the flight by continuing on to Hawaii after flying X miles toward Hawaii.
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But after flying X miles, if we wanted to turn back to Los Angeles, how much time would the flight back take? We would have to fly back the X miles that we had already come, only this time the speed would be the wind boosted 370 mph. So writing the equation for the time for the return trip would result in:
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T = X/370
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The point of no return is reached when the time to continue the flight to Hawaii equals the time it would take to fly back to Los Angeles. Since the two times are equal we can set the right sides of our two equations equal to each other as follows:
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(2553 - X)/250 = X/370
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Notice that this is a proportion. Proportions can be solved by cross-multiplying ... multiplying the numerator from one side by the denominator on the other side (doing it for both numerators) and then setting the two products equal. This is shown below:
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370*(2553 - X) = 250*X
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Do the distributed multiplication on the left side (multiply 370 times both terms in the parentheses) and the equation becomes:
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944610 - 370X = 250X
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Move the 370X to the other side by adding 370X to both sides:
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944610 - 370X + 370X = 250X + 370X
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On the left side the -370X and +370X cancel each other and on the right side the addition of the two terms results in 620X. So the equation is reduced to:
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944610 = 620X
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Solve for X by dividing both sides by 620 to get:
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1523.564516 = X
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And that's the answer that you were asked to find. When the flight is approximately 1523.6 miles out of Los Angeles, you might as well continue to Hawaii because it will take the same amount of time that it would to turn around and fly back to Los Angeles. (Note that you still have 2553 - 1523.6 = 1029.4 miles to go to reach Hawaii.)
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Hope this helps you to understand the problem.
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