1

Example 4 A road is to be constructed form city P to city Q as in the diagram below. The first part of this road PC lies along an existing road which costs $200,000 per km to renovate. The second part of this road CQ is new and costs $400,000 per km to construct. Where should C be chosen to minimize the cost of constructing this road?

Solution In the above picture we denote the distance from P to C by x, measured in km. The right triangle has sides of length 5-x and 3. By the Pythagorean

Theorem the length of its hypotenuse CQ is Let K denote the cost of constructing the road from P to Q. Then K is the sum of the cost of renovating the road PC plus the cost of constructing the road CQ:

with domain [0,5]. The problem is to find the value of x which minimizes K.

P B

Q

C

3 k m

5 k m

x 5-x

.)( 34x10x3x5 222

34x10x000400x000200K 2 ,,

2

To find the critical points, we set the derivative of K equal to zero:

since is not in the domain [0,5] of K. To find the minimum value of K we compare the values of K at the critical point and at the endpoints of the domain [0,5] of K:

35222552

22141010x

22x10x066x30x3025x10x434x10x

34x10x5x

41

34x10x

5x21

34x10x

5x000400000200

34x10x

5x000400000200

34x10x2

10x2000400000200

dxdK

0

2

2222

2

2

22

22

))(()(

and ,)(

)(

and ,,

,,,,

34x10x000400x000200K 2 ,,

3535

3

0002002300040050002005K

3813322340004003400040000002000K

23103926413851590653120004003500200

34351035000400350020035K 2

,,)(,)(,)(

,,,,)(,)(

,,,,,,)(,

)()(,)(,)(

Hence K has its minimum value at

34x10x000400x000200xK 2 ,,)(

km. 35x