A power cable must be laid from a power plant to a new factorylocated across a shallow river. The river is d feet wide and thefactory Is h feet downstream(PM) and g feet from the bank as shownin the figure. The cable costs $a per foot to lay under water, $bper foot to lay along the bank, and $c per foot to lay from thebank to the factory. What path should be taken to minimize the costand what is the minimum cos
t?
Let P,M, Q, R and F denote the points shown inthe diagram above. Note that P represents the location of the powerplant and F represents the location of the factory. The distancefrom P to Q can be found using the Pythagorean Theorem. You willneed to find the distance from R to F using the same method. Thedistance h = x + y + z is your constraint equation.
You may fill in the diagram with the followinginformation:
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 haveto:
Find a formula for the distance from P toQ.
Find a formula for the distance from R toF.
Use the formulas for the distances to writethe equation for the total cost of laying the cable. Call thisequation C(x, y, z).
4. 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, andz.
Give exact answers and decimalanswers rounded to two decimal places. When finding the decimalvalue of z, use the exact answers for x and y and not the roundedanswers. Round z to two decimal places. Explainwhy you should use the exact answers, as opposed to the roundedanswers, for x and y when finding the approximate answer forz.
5. Evaluate the minimum cost to complete theproject. You may use the two decimal approximate answers found inproblem 4. Round to the nearest cent.
