optimization -writing and solving the equations

[poppy]

Hi
Im trying to write an equation for the question below. Could someone please point me in the right direction with writing it?

An island is 4km from the nearest point p on the straight shoreline of a lake. if a person can row a boat at 3km/h and walk at 5km/h where should the boat be landed to arrive at a town 10km away is the least time?

I think the equation is y=(1/3)*(4^2+x^2)^(1/2)+((10-x)/5)

this doesn't look hard to differentiate but i cant seem to get the right answer-
in fact i get imaginary numbers

this is what i did

y=(1/3)*(16+x^2)^(1/2)-(1/5*x)+2
y'= -1/5+1/2*1/3(16+x^2)^(1/2)*2x
y'= -1/5+(2x/(6(16+x^2)^(1/2))
y'=0
1/5=x/(3(16+x^2)^(1/2))
3*(16+x^2)^(1/2)=5*x //square both sides
9*(16+x^2)=5*x^2
144+9x^2-5x^2=0
x= +/- 6i //this is obviously wrong as a person does not travel imaginary distances

Any help would be appreciated

 

[galactus]

We know that d=rt. Since the problem asks for least time, then we can use t=d/r.

Did you draw, or do you have, a diagram?. That would help visulaize.

\(\displaystyle t=\underbrace{\frac{\sqrt{x^{2}+16}}{3}}_{\text{time rowing}}+\overbrace{\frac{10-x}{5}}^{\text{time walking}}\)

This is what must be minimized.