Optimization -writing and solving the equations

[Poppynz]

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

Homework Statement

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?

Homework Equations

The Attempt at a Solution

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 can't 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

 

[tiny-tim]

Welcome to PF!

Hi Poppynz! Welcome to PF!

(have a square-root: √ and try using the X² tag just above the Reply box )

… 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

oops! … you forgot to square the 5, so you got 5 - 9 instead of 25 - 9

Try again! ​

 

[Poppynz]

haha such a silly mistake thanks :)