need help...
Hello, bobby,
I label the island I.
The total distance is d = IC + CD
The total energy is e = 2*IC + CD
Let BC = x
Then you can calculate the energy by:
$\displaystyle e(x) = 2 \cdot \sqrt{5^2+x^2} + (13-x)$
You'll get the minimum if e'(x) = 0. So you need first the derivative of e (use chain rule!):
$\displaystyle e'(x) = 2 \cdot \frac{1}{2}\cdot \left( 5^2+x^2 \right)^{-\frac{1}{2}} \cdot 2x -1$
Now e'(x) = 0:
$\displaystyle 2 \cdot \frac{1}{2}\cdot \left( 5^2+x^2 \right)^{-\frac{1}{2}} \cdot 2x -1=0$
$\displaystyle \frac{2x}{\sqrt{25+x^2}} =1$. Multiply both sides by the denominator and afterwards square both sides:
$\displaystyle 4x^2 = 25+x^2$. Solve for x. You'll get 2 values for x. The negative one is senseless in the described situation.
I got $\displaystyle x=\frac{5}{3}\cdot \sqrt{3}\approx 2.887 \text{ km}$
EB