Hi I am stuck on a optimization problem so I am here for some assistance. Heres the question:

Simon Suffers an injury at a campsite on the bank of a canal that is 6√2 km wide. If he is able to paddle his kayak at 5 km/h and pedal a bicycle at 15 km/h along the opposite bank, where should he land on the opposite bank to reach the hospital in minimum time?

here is a diagram that comes with the question:

So out of this question I've determined this:

Distance = Velocity * time <- this can be re-arranged to this -> time = Distance / Velocity

distance on kayak: √((6√2)^2 + x^2)

distance on land: (4 - x)

so my formula is this:

Time(x) = (1/15)(4 - x) + (1/5)(√(x^2 + 72))

I differentiated and got this:

Time'(x) = [ (x) / (5 * √(x^2 + 72) ) ] - (1/15)

Now the next step I do is I set my Time'(x) = 0 and solve for x but I feel that I am getting a bad answer. Heres what I have:

t’(x) = 0

(x/(5sqrt(x^2 + 72)) – (1/15) = 0

(x/(5sqrt(x^2 + 72)) = (1/15)

(x)^2 = (1/15)^2(5sqrt(x^2 + 72)^2

x^2 = (1/225)(25)(x^2 + 72)

(x^2) * 225 = ((1/225)(25)(x^2 + 72))*225

(225x^2)/25 = ((5625)(225x^2 + 405000))/25

9x^2 = (225)(9x^2 +16200)

9x^2 = 2025x^2 + 3645000

9x^2 - 2025x^2 = 3645000

-2016x^2 - 3645000

x^2 = -1808.35714x^2

So I cant continue this because as you can see 3645000 does not divide nicely with -2016 and I cant take a square root of a negative number.

If someone could point out where the mistake is that would be GREATLY !! appreciated!!!

Thanks!!!