# Thread: optimization -writing and solving the equations

1. ## optimization -writing and solving the equations

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

2. Originally Posted by 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
Better use of brackets needed:

$\displaystyle y'= -\frac{1}{5}+\left(\frac{1}{2}\right)\;\left(\frac{ 1}{3}\right)(16+x^2)^{-1/2}(2x)=0$

Where $\displaystyle x$ is the distance of land fall measured along the shore line from the nearest point on the shore line to the island, and $\displaystyle y$ is the time to make the journey.

So:

$\displaystyle 3(16+x^2)^{1/2}=5x$

squaring:

$\displaystyle 9\times 16+9\times x^2=25x^2$

or:

$\displaystyle 16x^2=9 \times 16$

CB

3. ## Thanks

haha thankyou for pointing out my algebra mistakes seems so obvious now ...