Business Problem: Cost of construction and optimum route: Towns A and B are on opposite sides of a ridge. A road with a tunnel with a tunnel under the ridge is to be built connecting them. The cost of construction is $0.5 million per mile on the level ground beside the ridge, and $2.0 million per mile for a tunnel and road under the ridge. Find the lowest cost route for the road.

 We can describe the situation as in the diagram. Let H be the length of the tunnel portion, 5 - L be the length of the level portion beside the ridge. We can express the total cost in terms of the lengths 5 - L and H and the cost per mile of each type of construction. Using millions of dollars:

We want to minimize the cost. In order to do so, we need to express it in terms of a single variable. H, L and the 0.5 mile distance are related by the Pythagorean Theorem:

.

With this result, we can express the cost entirely in terms of the length L:

Now we can differentiate with respect to L and find the value of L which minimizes the cost.

In general,

At the minimum,

Collecting the dependence on L_min on one side, we get

We can square both sides to get

, or

Solving, we get

, or

Hence,

The corresponding cost is

For comparison, when L is 0.14, the cost is 3.468 million,

and when L is 0.12, the cost is 3.468 million.