Writing in Mathematics

Question

a public works department is to construct a new road between towns A and B. Town A lies on an abandoned road running east-west. Town B lies 40 miles east of Town A and 20 miles north of the road. An engineer proposes that the road be constructed by restoring a section of the old road leaving Town A and joining it to an entirely new section of road at a point C to be determined, connecting that point with town B as shown in the figure below. If the cost of restoring the old road is $200,000 per mile and the cost of new construction is $400,000 per mile, determine the function describing the total cost. Also graph the cost function and estimate how much of the old road should be restored to minimize the department's cost.

Answer 1

Draw a diagram.

 

Point D is on the old road directly south of B.

Assume x is the distance from A to C. The distance from C to D is 40 - x. This distance is one leg of a right triangle.

The hypotenuse is √(400 +(40 - x)²)

 

The cost to redo the old road: 200000x

The cost of the new road:       400000(√(400 + (40 - x)²))

 

Total cost: 200000x + 400000(√(400 + (40 - x)²)