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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
