A power cable must be laid from a power plant to a new factory located across a

Question

A power cable must be laid from a power plant to a new factory located across a shallow river. The river is d feet wide and the factory Is h feet downstream(PM) and g feet from the bank as shown in the figure. The cable costs $a per foot to lay under water, $b per foot to lay along the bank, and $c per foot to lay from the bank to the factory. What path should be taken to minimize the cost and what is the minimum cost? F Let P,M, Q, R and F denote the points shown in the diagram above. Note that P represents the location of the power plant and F represents the location of the factory. The distance from P to Q can be found using the Pythagorean Theorem. You will need to find the distance from R to F using the same method. The distance h = x + y + z is your constraint equation. You may fill in the diagram with the following information: d = 50 ft. and g = 50 ft. h = 100 ft. a = $500, b=$100, and c = $300 In order to solve this problem, you will have to: 1. (2 pts) Find a formula for the distance from P to Q. 2. (2 pts) Find a formula for the distance from R to F. 3. (5 pts) Use the formulas for the distances to write the equation for the total cost of laying the cable. Call this equation C(x, y, z). 4. (6 pts) Use the Lagrange multiplier technique, subject to the constraint: x + y + z = h, to find the values of x, y, and z which will yield a minimum value of this equation. Make sure to give the values of x, y, and z. Give exact answers and decimal answers rounded to two decimal places. When finding the decimal value of z, use the exact answers for x and y and not the rounded answers. Round z to two decimal places. Explain why you should use the exact answers, as opposed to the rounded answers, for x and y when finding the approximate answer for z. 5. (3pts) Evaluate the minimum cost to complete the project. You may use the two decimal approximate answers found in problem 4. Round to the nearest cent.

Explanation / Answer

under river distance = sqrt[d^2 + h^2] total distance = sqrt[d^2 + h^2] + g total cost = sqrt[d^2 + h^2]*a + gb

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