Question 643926: At noon a cargo plane leaves mchare airport and heads east at 180 mph. It's destination is jamesville, 500 miles away.at 1:00 pm a her takes off from mchare and lies east after the cargo plane at 450 mph. At what time will the jet overtake the cargo plane?
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! Let  be the time in hours after the cargo plane leaves at noon.
(At 1:00 PM, t=1.00, at 2:00 PM, t=2.0, at 2:30 PM, t=2.5, and so on).
Assuming that the cargo plane continues flying to jamesville at 180 miles per hour, at 1:PM, the cargo plane will have been flying for 1 hout; will have covered 180 miles, and will be at 180 miles from mchare airport.
At 2:PM, with 2 hours on the air, the cargo plane will be at 360 miles from mchare airport.
At some point after that, but before 3:00PM, the cargo plane will have covered the 500 miles to Jamesville.
We do not know where the cargo plane was before noon, or what it would do after reaching jamesville.
The distance between the cargo plane and mchare airpot is
 for between time zero and the time when the cargo plane arrives at jamesville.
Assuming that the jet plane continues flying after the cargo plane, the distance between the jetplane and mchare airpot is
 for between  and the time when the jet plane catches up with the cargo plane.
We do not know what the jey plane is doing before 1:00PM, or after it catches up with the cargo plane.
Since they are both flying on the same trajectory, the jet plave will overtake the cargo plane when they are both at the same distance form mchare.
That will happen when  ,
meaning 
We need to solve that equation for
First we apply the distributive property to the left side of the equation:
--> 
Next, we subrtract  --> 
-->  -->  -->  -->  --> 
Finally, we divide both sides of the equation by  :
--> -->  -->  --> 
We know that 1 hout is 60 minutes,  of an hour is 20 minutes and  of an hour is 40 minutes, so
 hours is 1 hour and 40 minutes, and since is the time counting from noon, then the jet plane overtakes the cargo plane at 1:40 PM.
Verification:
At 1:40 PM, .
The cargo plane has been flying for  hours at 180 mph, and has covered
 miles = 300 miles
The jet plane, that left at 1:00 PM ( ),
had been flying for hour at 450 mph, and has covered
 miles = 300 miles.
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