Word problem in calculus.?

Question

Rosie O'Donnell is swimming 50 meters from the shore. When she looks up she sees Tom Cruise standing at water's edge 100 meters down the beach from where Rosie is swimming. If you have watched Rosie's television show, you know that she has a thing for Tom Cruise. Rosie's top speed swimming is one meter per second. Her top speed running is 4 meters per second. What point on the beach should she swim for in order to minimize her time to reach Tom Cruise?

Answer 1

Let x be the distance down the beach that she should aim to swim to shore. So she has to run 100-x meters.
She swims sqrt(50^2 + x^2) meters.

Total time, t = sqrt(50^2 + x^2) + (100-x)/4

dt/dx = x / sqrt(50^2 + x^2) - 1/4
This is zero when
4x = sqrt(50^2 + x^2)
15x^2 = 50^2
x = 12.9 meters
Her total time would be 73.4 seconds.

Answer 2

The situation is as sketched below

Cruise -
-
- - 100m
x
- -
-
-
- Rosie -
--------
500m

She has to swim to point x at 1 m/s and the run from x to Cruise at 4 m/s. Let h be the distance from x to Cruise. The initial distance from Rosie to x is D = sqrt(2500 + (100 - h)^2). To reach x, it'll take Rosie that time t1 = D/1 = D. And to reach Cruise running from x, it'll take Rosie the time t2 = h/4. So, the total time t to reach Cruise is t = t1 + t2 = sqrt((2500 + (100 - h)^2)) + h/4 We have to find h so that t is minimum.

Differentiating, we get dh/dt = (h -100)/ sqrt((2500 + (100 - h)^2)) + 1/4. Setting dh/dt = 0, we get

4(100 - h) = sqrt(2500 + (100 -h )^2)

Then, you solve this equation, check that it's a point of minimum and have the solution.

Answer 3

Um, a straight line between her and Tom, while compensating for tide and whatever.