Please follow the instructions showing every step and thanks! ------------------

Question

Please follow the instructions showing every step and thanks!

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On the same side of a straight river are two towns, and the townspeople want to build a pumping station, S. The pumping station is to be at the river's edge with pipes extending straight to the two towns as shown in the figure below. Let A = 6 Express the total length of the pipes in terms of x only. Find L'(x) = Show your work for L'(x) and simplify completely Where should the pumping station be located to minimize the total length of pipe? Give an exact answer. Show the algebra needed to find the exact value of the critical point. Show why the critical point is the global minimum

Explanation / Answer

1. A = x/sqrt(x^4+2)

A' =(sqrt(x^4+2) - 2x^4/sqrt(x^4+2)/(x^4+2) =

(x^4+2 - 2x^4)/(x^4+2)^(3/2) =

(2-x^4)/(x^4+2)^(3/2)

This equals 0 when 2-x^4 = 0, or x = 2^(1/4)

We don't need to calculate the second derivative,as the denominator of the first derivative is positive and the numerator, 2-x^4 is positive for x < 2^(1/4) and negative for x > 2^(1/4)

c) The critical point, x = 2^(1/4), is a maximum. The derivative is positive for x on (0, 2^(1/4) and negative on (2^(1/4), inf), so this is a global maximum.

d) The dimensions are x = 2^(1/4) and y = 1/sqrt((2^(1/4)^4 + 2) = 1/sqrt(2+2) = 1/sqrt(4) = 1/2

Thus, the maximum area is 2^(1/4) * 1/2 = 2^(1/4)/2 or 2^(-3/4) = 0.594603557501361

2. a) as the distance along the river from town A to S is x, the distance along the river to the 90 degree turn is A-x = 6-x

Thus, the length of the pipe = sqrt(1^2+x^2) + sqrt(6^2+(6-x)^2) = sqrt(1+x^2) + sqrt(36+(6-x)^2)

b) L' = x/sqrt(1+x^2) + (x-6)/sqrt(36+(6-x)^2) =

(x sqrt(36+(6-x)^2) + (x-6) sqrt(1+x^2))/sqrt((1+x^2)*(36+(x-6)^2))

c) This equals 0 when x sqrt(36+(6-x)^2) + (x-6) sqrt(1+x^2) = 0, or

x sqrt(36+(6-x)^2) = (6-x) sqrt(1+x^2)

Squaring both sides,

x^2(72-12x+x^2) = (36-12x+x^2)(1+x^2)

72x^2 - 12x^3 + x^4 = 36-12x+37x^2-12x^3+x^4

35x^2 + 12x - 36 = 0

(7x - 6)(5x + 6) = 0

x = 6/7, x = -6/5

We only need to consider x > 0, so the solution is x = 6/7

To see that this is a minimum, note that this is the unique 0 for x > 0, and when x = 0, the total distance is 1 + sqrt(72) =

9.48528137423857, and the distance at x = 6 is sqrt(37) + 6 = 12.0827625302982, while the distance at 6/7 is

sqrt(1+(6/7)^2) + sqrt((6-6/7)^2 + 36) = sqrt(1+(6/7)^2) + sqrt((36/7)^2+36) = sqrt(1+(6/7)^2) + 6(sqrt(6/7)^2+1) =

7 sqrt((6/7)^2+1) = sqrt(85) = 9.21954445729289 is less.

Note: the easy way to solve this geometrically is to place the town 1 mile North of the river instead. Then, the point S is in a straight line from Town A to Town B.

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