# Question with Optimization

There is an island which is 3 km west and 2 km north of a power station (The x axis which the power station is on is made of land, the y axis is not). If it costs 25,000 dollars per km to lay a power line underground an 50,000 dollars per km to lay a power line under water, find the most cost-efficient way of distributing land vs water power lines and state the minimal total cost.

I need help with this and have no idea of how to start. My idea (which is probably wrong) is to find the hypotenuse and multiply by 50,000 but that sounds too easy... can I get some help please.

The picture for this situation looks like this:

The cost function is that given in the comment attached to angryavian's answer. A discussion for a similar optimization problem (of a very commonly-assigned sort) can be found here. Differentiating the cost function to find its critical point gives us

$$\frac{dC}{dx} \ = \ 25000 \ \left( \ 1 \ + \ 2 \ \cdot \frac{1}{2} \ [ \ 2^2 \ + \ (3-x)^2 \ ]^{-1/2} \ \cdot \ 2 \ (3-x) \ \cdot (-1) \right) \ = \ 0$$

$$\Rightarrow \ \ \frac{2 \ (3-x)}{\sqrt{2^2 \ + \ (3-x)^2}} \ = \ 1 \ \ \Rightarrow \ \ 3 \ - \ x \ = \ \sqrt{\frac{4}{3}} \ \ \Rightarrow \ \ x \ \approx \ 1.85 \ \text{km.} \ \ .$$

This makes the minimal total cost

$$C(1.85) \ \approx \ 25000 \ \cdot \ 1.85 \ + \ 50000 \ \sqrt{4 \ + \ (3 -1.85)^2} \ \approx \ 161,700 \ \text{dollars} \ \ .$$

Hint: Distance-wise, the hypotenuse is the shortest distance, as you noted. However, that path is entirely underwater, which is very expensive. The other extreme would be to lay 3 km of land line west, and then 2km of underwater line north, but that has a large distance and may still cost too much. The optimal way is probably somewhere in the middle: lay some land line to the west for $x$ kilometers, and then take the straight-line distance $\sqrt{2^2+(3-x)^2}$ to the island from there.

• Hi @angryivan I did that and said that x was 2 since its better to travel most on land. So I ended up getting my answer as 161,808 dollars. Is that correct? – Panthy Jun 23 '14 at 0:49
• @Panthy I did not get $x=2$. Did you take the derivative of $25000x+50000\sqrt{2^2+(3-x)^2}$? – angryavian Jun 23 '14 at 1:23
• oh i didnt do that – Panthy Jun 23 '14 at 1:43