(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Albany is 12km north of Rochedale. Bells Creek is 5km west of Albany. The road is to be repaired and repairs cost $96 000 per km. The cost of laying a new road is $120 000 per km.

It has been decided to repair the old road from Rochedale as far as point K, and then to build a new road from K to Bells Creek. Find the length of the new road such that the cost of the whole project is a minimum.

2. Relevant equations

3. The attempt at a solution

Let C = total cost

Let the distance from point K to Bells Creek = x

Let the distance from Rochedale to point K = y

Let the distance from point K to Albany = b

Thus the y can be written in terms of b using the 12km from the initial question.

y = 12 - b

C = 120000x + 96000y

Substitute y=12 - b into the equation

C = 120000x - 96000b + 1152000

Then there is a right angle triangle from point K to Albany to Bells Creek, with the hypotenuse being the new road and the variable x defined above.

Thus the side b (dsitance from point K to Albany) can be written in terms of x

b = √(x^2 - 25)

Then substitute the value for b into the total cost equation.

C = 120 000x - 96000(√(x^2 - 25)) + 1152000

Not really sure where to go from here. I have done lots of problems and understand the concept of finding the minimum or maximum by making the derivative equal to zero but am not sure how to go about either simplifying or deriving the middle term because of the square root sign.

Any help would be greatly appreciated.

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