Question

Brooke is located 5 miles out at sea from a straight shoreline in her kayak. She wants to make it to the taco truck on the beach for lunch, which is 6 miles from point A on the shore (see picture) Brook can paddle 2 miles per hour and walk 4 miles per hour. If she paddles along a straight line to shore, find an equation for the total time it will take Brooke to get to lunch. Your equation will depend on where Brooke beaches her kayak. Where should she beach the kayak to eat as soon as possible?

Answer 1

Answer:

x  =  1,2909  ml   (point where Brooke beaches her kayack)

Step-by-step explanation:

I did not find the indicated picture.  I design one  (See picture in Annex)

The route to follow by Brooke  is  x₁   +  x₂ and as we need to express total time.

s  (distance)  =  velocity* time    then     time t  =  distance / velocity

x₁  = √ (5)² + x²        ⇒  x₁  = √( 25 + x² )

and   x₂   =  (  6  -  x  )

x₁  = distance in kayack     ( speed 2 ml/h)

x₂  = distance walking       ( speed  4 ml/h)

Then total time as a function of x is

t(x)   =  √( 25 + x² ) / 2     +   (  6  -  x  ) / 4

Taking derivatives on both sides of the equation we get

t´(x)   =  2x* (2) / 4 *√( 25 + x² )  - 1/4

t´(x)   =  0       2x* (2) / 4 *√( 25 + x² )  - 1/4  = 0      x/√( 25 + x² ) - 1/4  = 0

x/√( 25 + x² )  = 1/4

x    =   √( 25 + x² ) /4

x²   =  (25 + x² ) /16

16*x²   = 25  +  x²

15x²   =  25          x  =  1,2909  ml   (point where Brooke beaches her kayack)

Then  walking distance is   x₂   =  6  - 1.2909    x₂  = 4.71 ml