need help...

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- Nov 5th 2006, 09:58 PMbobby77minimum value
need help...

- Nov 5th 2006, 10:31 PMearboth
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