The State wants to build a power station along an East-West highway to supply power
to four small towns, A, B, C, & D located along the highway. Town C is 100 miles West of D and 85 miles East of B. Town A is 30 miles West of B. The cost of connecting the power station to any town in $1000 per mile of cable. The total budget cannot exceed $230,000. Where can the power station be located? You must provide a separate cable connection from the power station to each town.

For the total budget not to exceed $230,000 the distance from town A to the power plant should be 147.5 miles

Step-by-step explanation:

We are given the cost of connection of power to a town as $1,000.00 per mile of cable

Location of the towns as

A = 30 miles West of B

C = 85 miles  East of B

C = 100 miles west of D

Therefore let the location of the power station be Z such that the total cost of supplying power to the four towns = $230,000

And let the distances of the town from the power station be

Z - A = a miles

Z - B = b miles

Z - C = c miles and

Z - D = d miles

Therefore, (a + b + c + d) × 1000 = $230,000

Hence, (a + b + c + d) = 230 miles

Since the towns are;

A = Start

B = 30 miles

C = 115 miles

D = 215 miles

4·Z - (A + B + C + D) = 230 where A = 0

4·Z - (B + C + D) = 230

4·Z - (30 + 115 + 215) = 230

4·Z = 230 + 360 = 590

Z = 147.5 miles

The location of the power plant should be at 147.5 miles  from town A.