The Least Expensive Cable Better Cable Company must provide service to a customer whose house is located 2 miles from the main highway. The nearest connection box for the cable is located 5 miles down the highway from the customers’ driveway. The installation cost is $14 per mile for any cable that is laid from the house to the highway. (The cable may be laid along the driveway to the house or across the field). The cost is $10 per mile when the cable is laid along the highway. Determine where the cable should be laid so that the installation cost as low as possible.

Paying for step by step help

Let x be the distance (in miles) between the customer's driveway and the point where the cable should be laid on the highway.

Length of cable along the highway : 5 - x
Cost : 10.( 5 - x ) dollars

Length of cable along the countryside from the highway to the house : √(4+x²)
(use the Pythagorean theorem to prove this)
Cost : 14.√(4+x²)

Total cost :
f(x) = 14.√(4+x²) + 10.( 5 - x )

Study that function of x, while 0<x<5.

f '(x) = 14x/√(4+x²) - 10
f '(x) = 0 iff x = 5/√(6)

That's the point where the cable should be installed.

Get the final (and minimum) cost by evaluating f (5/√6) = 50 + 8√6 ~ 69.6 dollars

If I do no mistake....