Expert: Paul Klarreich - 1/17/2010

Question
You have just arrived at the beach—in fact, you have just parked your car in a lot 100 m from
the shore. You need to deliver a cooler full of drinks to your friends who are frolicking at the
shoreline 300mdown the beach fromyour parking spot (see the picture, below).
Since the cooler is heavy you, want to get to your friends as quickly as possible. You can travel
across the sandy beach, but it’s slow going. Luckily, there is a boardwalk that runs the parallel
to the shoreline. You can travel along the boardwalk three times as fast as you can travel across
the sandy beach.
(a) What route gets you to your friends in the least amount of time?


Answer
Questioner: Sarah
Country: Canada
Category: Calculus
Private: Yes
Subject: Minimizing time
Question: You have just arrived at the beach—in fact, you have just parked your car in a lot 100 m from the shore. You need to deliver a cooler full of drinks to your friends who are frolicking at the shoreline 300m down the beach from your parking spot (see the picture, below).

>>>>>>>>>>>>>>> What picture below?

Since the cooler is heavy you, want to get to your friends as quickly as possible. You can travel across the sandy beach, but it’s slow going. Luckily, there is a boardwalk that runs the parallel to the shoreline. You can travel along the boardwalk three times as fast as you can travel across the sandy beach.
(a) What route gets you to your friends in the least amount of time?
..............................
let x = BC in the attached picture (If I remember to.)

You walk AB, then BD across the sand.

Assume you walk 3 m / sec on AB, 1 m/sec on BD

AB = 300-x
time(AB) = (300-x)/3

BD = sqrt(x^2 + 100^2)
time(BD) = sqrt(x^2 + 100^2)  {over 1, of course}

T = total time

T = (300-x)/3 + sqrt(x^2 + 100^2)

T' = -1/3 + x/sqrt(x^2 + 100^2)


-1/3 + x/sqrt(x^2 + 100^2) = 0

3x/sqrt(x^2 + 100^2) = 1

3x = sqrt(x^2 + 100^2)

9x^2 = x^2 + 100^2

8x^2 = 100^2

x^2 = 100^2/8

x = 100 / sqrt(8) = 50/sqrt(2)