SOLUTION: you are standing on the west bank of a calm river which is 3km wide. your friends are waiting for you on the east side of the river 10 km south of your location. you will take a ca

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Question 1097839: you are standing on the west bank of a calm river which is 3km wide. your friends are waiting for you on the east side of the river 10 km south of your location. you will take a canoe across the river to some point P, and then run the remaining distance to join with your friends. if you can canoe at 4.5 km/h and run at 7 km/h, what is the shortest amount of time that it will take to be reunited with your friends?
Answer by greenestamps(13198) About Me  (Show Source):
You can put this solution on YOUR website!

Let A be the point where you are standing; let B be the point on the east bank directly opposite of where you are standing: let C be the point on the east bank where you will beach your canoe and start running; and let D be the point where your friends are waiting for you.

You are looking for the minimum total time required to reach your friends. That means you need to form an expression in terms of some variable for the total time and find the minimum value of that expression. Finding the minimum value means finding where the derivative of the total time expression is equal to zero.

Let length BC be x; then CD is (10-x).

AC is the hypotenuse of right triangle ABC with legs of length 3 and x; you will be rowing that distance at a rate of 4.5 km/h. CD is the distance you will be running at 7 km/h. The total time required is

sqrt%28x%5E2%2B9%29%2F4.5+%2B+%2810-x%29%2F7
or
%28x%5E2%2B9%29%5E%281%2F2%29%2F4.5+%2B+%2810-x%29%2F7

The derivative of this expression is

%281%2F2%29%28x%5E2%2B9%29%5E%28-1%2F2%29%2A%282x%29%2F4.5+-+1%2F7

Setting this expression equal to 0 and solving for x...
x%2F%284.5%2Asqrt%28x%5E2%2B9%29%29+=+1%2F7
7x+=+4.5%2Asqrt%28x%5E2%2B9%29
49x%5E2+=+20.25%28x%5E2%2B9%29
49x%5E2+=+20.25x%5E2%2B182.25
28.75x%5E2+=+182.25
x%5E2+=+182.25%2F28.75
x+=+sqrt%28182.25%2F28.75%29+=+2.517763 approximately

Substituting this value into the expression for the amount of time required (I won't show the arithmetic!) gives a minimum time of about 1.93923 hours.

This result was verified by using a graphing calculator to find the minimum value of the expression for the total amount of time required.