Grades 9-10 Video Solutions 2014-Levels 9&10 Video Solutions 2014 problem17

Levels 9&10 Video Solutions 2014 problem17

Video Transcription

Question number 17. Anne walked 8 kilometers with a velocity of 4 kilometers per hour. Now she will run for some time with a velocity of 8  kilometers per hour. How long does she have to run in order to have an overall average velocity of 5 kilometers per hour? Let's recall the definition of  average velocity. We know that 5 kilometers per hour being the average velocity has to be the total distance  divided by the total time. And with that we have two legs of the trip. First Anne is walking. She is walking for 8 kilometers and she's doing that in the  amount of time that is 8 kilometers divided by her velocity of 4 kilometers per hour and that comes out to 2 hours. Then she is running. We don't know how  long for what time she is running or for how long so let's say she has with her velocity of 8 kilometers per hour she is traveled 8 kilometers per hour at  some time t and exactly in t hours. Okay and then we have the right-hand side of our equation here. We add 8 kilometers to 8 kilometers per hour multiplied by  t hours and we divide all that by the time which is 2 hours plus t hours and we see that the units here give us exactly kilometers per hour on the  right-hand side in blue so let's just dispense with the units for now and solve the equation here. So for t the equation 5 is equal to we have 8 plus  8t divided by 2 plus t with the units gone and so what we should do is we should say move everything over to one side of the equation so we have here  negative 5 plus 8 plus 8t divided by 2 plus t. Now we can look for a common denominator negative 5 times 2 plus t plus 8 plus 8t divided by 2 plus t  that's the common denominator and so what we have is negative 5 times 2 plus 8 so that's negative 10 plus 8 negative 2 and we have negative 5 times t plus 8t  plus 3t divided by 2 plus t so the solution is this is equal to 0 provided that t is equal to two-thirds of an hour okay and now two-thirds of an hour  two-thirds of one hour is two-thirds of 60 minutes and we can compute that as 40 minutes that is exactly 40 minutes so that's the value of t required here so  the average velocity comes out to five kilometers per hour two-thirds of an hour or 40 minutes and so that's how long n has to travel at eight kilometers  per hour to make that average and so the answer here is E

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