1 Example 4 A road is to be constructed form city P to city Q as in the diagram below. The first...
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Transcript of 1 Example 4 A road is to be constructed form city P to city Q as in the diagram below. The first...
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