can you help me with this calculus problem? its easy i just forgot how to solve it?

Question

A variation of a triathlon competition has a contestant swimming from point A, on one shore of the lake to point C on the opposite parallel shore, then running to the finish, B, futher along the lakeshore. the lake is 4km wide and the finish line ( point B) is 10 km down the lake from the start (POINT A) . If the contestant can swim at 2 km/h and run at 10km/h, determine the point C that will minimize the total time for the race.

Answer 1

Let AC^2 = x^2+4^2. Then BC = 10-x.
Let t1 be the swimming time, t2 be the running time, and t be the total time.

By distance = velocity x time,
t = t1+t2 = AC/2 + BC/10 = √(x^2+4^2)/2 + (10-x)/10

From dt/dx = x/√(x^2+4^2)/2 - 1/10 = 0, we solve for x,
25x^2 = x^2+4^2
x = 0.8165 km

Therefore, point C is 0.8165 km down the lake from the start but on the opposite shore of the lake.

Answer 2

The swimming distance is S = √[W^2 + AC^2] W is lake width, AC the distance from A to C along the shore. The running distance is L = 10km - AC. The total time is then S/vs + L/vr where vs = swim velocity and vr = run velocity. Take the derivative and set to 0 to find the value of AC for minimum time.

NOTE: there are two errors in sahsjing's solution. The derivative of √[x^2+4^2]/2 = x/2√[x^2+4^2], and he forgot to square the 10 on the right side. The real answer is √[64/96] = .816km NOTE AGAIN: I guess he corrected it while I was writing this.