Can someone help me find the correct answer to the following question? My Notes

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Can someone help me find the correct answer to the following question?

My Notes 14. 015 points | Previous Answers SCalcET7 4.7.050. An oil refinery is located 1 km north of the north bank of a straight river that is 2 km wide. A pipeline is to be constructed from the refinery to storage tanks located on the south bank of the river 5 km east of the refinery. The cost of laying pipe is $200,000/km over land to a point P on the north bank and $400,000/km under the river to the tanks. To minimize the cost of the pipeline, how far downriver from the refinery should the point P be located? (Round your answer to two decimal places.) 3.84 km Need Help?Read It Chat About It

Explanation / Answer

Well first you write out the equation that you want to minimize. In this case you want to minimize the cost. So you want to minimize:

C = 200000*l + 400000*r

where l is the number of kilometers of piping over land and r is the number of kilometers of piping over water.

The relationship between r and l (constraint) is now:

r = sqrt((3 - l)^2 + 2^2)
r = sqrt(13 - 6l + l^2)

That may look a little confusing. But r in this case is the hypotenuse of a triangle (a^2 + b^2 = c^2) - 2 is one side of the triangle (the one that goes across the river) and (3-l) is the side of the triangle equal to the distance along the river between P and the storage tanks. Wish I could draw it out.

Now we substitute for r from the constraint into the minimization problem:

C = 200000*l + 400000*sqrt(13 - 6l + l^2)

We want to find the value of l that makes C as small as possible. We do this by first differentiating C in terms of l:

DC/dl = 200000 + 200000*(-6 + 2l)*(13-6l+l^2)^(-1/2)

I hope you're allowed to use a calculator, now. Set that equal to 0. There is only one critical point, l = 1.8453 km, which is the answer.

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