You can run at a speed of 8 mph and swim at a speed of 4 mph and are located on the shore, 5 miles east of an island that is 1 mile north of the shoreline.

Question

How far (in mi) should you run west to minimize the time needed to reach the island?

Answer 1

If you make a sketch, it might look like this:

"x" is the distance you run and "d" is the distance you swim. You can use the Pythagorean Theorem to solve for "d":

1² + (5 - x)² = d²

d = √(1 + (25 - 10x + x²)

d = √(x² - 10x + 26) = (x² - 10x + 26)^1/2

time = distance/rate so the time running (in hours) is x/8 and the time swimming (in hours) is (x² - 10x + 26)^1/2/4 so the total time equation is:

t(x) = x/8 + 1/4(x² - 10x + 26)^1/2

To minimize, take the derivative (using the chain rule) and set it equal to zero.

t'(x) = 1/8 + 1/8(x² - 10x + 26)^-1/2(2x - 10)

0 = 1/8(1 + (2x - 10)/(x² - 10x + 26)^1/2

0 = 1 + (2x - 10)/(x² - 10x + 26)^1/2

-1 = (2x - 10)/(x² - 10x + 26)^1/2

-1(x² - 10x + 26)^1/2 = 2x - 10

[-1(x² - 10x + 26)^1/2]² = (2x - 10)²

x² - 10x + 26 = 4x² - 40x + 100

3x² - 30x + 74 = 0

(using the quadratic formula) x = 4.423 and x = 5.577 (the 2nd answer doesn't make sense in this case so throw it out)

So you should run 4.423 miles