Hagrid is 3 offshore in a boat transporting some first years to Hogwarts

Question

Hagrid is 3 offshore in a boat transporting some first years to Hogwarts, located 6 down a straight shoreline from a point near the boat. He can row 2 mph and they can walk 5 mph. Where should he land the boat to reach Hogwarts in the least amount of time?

Answer 1

We shall now find the minimum time. Apologies if there was any confusion on the video. When I said swim, I really meant to say "row".

In the video, I earlier stated that the total time combined (with changing x with sqrt(u^2 + 9)) is:

T = 1.2 - (u / 5) + 0.5sqrt(u^2 + 9).

Remember, you can find the minimum of t by taking the derivative, and see that IF at when t'(u) = 0, t' changes from negative to positive, it's a minimum.

d/du( T(u) ) = - 0.2 + u / ( 2 *sqrt(u^2 + 9) ).

T ' (u) = 0 when (after algebraically solving for u) u = 6 / sqrt(21).

Checking the derivative ensures that when u < 6 / sqrt21, t'(u) is negative, and when u > 6 / sqrt21, t'(u) is positive. Therefore, Hagrid can land the boat at u = 6/sqrt21 miles to ensure the smallest time possible.