Question

A business owner needs to run a gas line from his business to a gas main as shown in the accompanying diagram. The main is 30-ft down the 12-ft wide driveway and on the opposite side. A plumber charges $4 per foot alongside the driveway and $5 per foot for underneath the driveway.

a) What will be the cost if the plumber runs the gas line entirely under the driveway along the diagonal of the 30-ft by 12-ft rectangle? b) What will be the cost if the plumber runs the gas line 30-ft alongside the driveway and then 12-ft straight across? c) The plumber claims that he can do the job for $160 by going alongside the driveway for some distance and then going under the drive diagonally to the terminal. Find x, the distance alongside the driveway. d) Write the cost as a function of x and sketch the graph of the function. e) Use the minimum feature of a graphing calculator to find the approximate value for x that will minimize the cost. f) What is the minimum cost (to the nearest cent) for which the job can be done?

Answer 1

.

a. As per the Pythagoras theorem, if the diagonal of the rectangle is d ft., then d² = 30² +12² = 900+144 = 1044 so that d = ?1044 = 32.31 ft. (approximately, on rounding off to 2 decimal places). Thus, if the plumber runs the gas line entirely under the driveway along the diagonal, then the cost will be 32.31* $ 5 = $ 161.55.

b. If the plumber runs the gas line 30-ft alongside the driveway and then 12-ft straight across, then the cost will be (30+12)*$ 4 = 42*$ 4 = $ 168.

c. If the plumber runs the gas line alongside the driveway for x ft. and then goes under the driveway diagonally to the terminal, then the length , d, of the diagonal, is given by d² = 12²+(30-x)² = 144 +900 -60x +x² = x² -60x + 1044 so that d = (x² -60x + 1044)^1/2. The cost of x ft. of gas line alongside the driveway is $ 4x and the cost of d =(x² -60x + 1044)^1/2 ft. of gas line under the driveway is $ 5(x² -60x + 1044)^1/2. Since the total cost is $ 160, we have 4x +5(x² -60x + 1044)^1/2 = 160 or, 5(x² -60x + 1044)^1/2 = 160-4x. On squaring both the sides, we get 25(x² -60x + 1044) = (160-4x)² or, 25x² -1500x + 26100 =25600 -1280x +16x² or, 9x²- 220x+500 = 0. Now, on using the quadratic formula, we get x = [220±?{(-220)²-4*9*500}]/2*9 = [220±?(48400-18000)]/18 = (220±?30400)/18 = (220±174.36)/18 = 2.54 ft. or, 21.91 ft. (approximately, on rounding off to 2 decimal places).

d. The cost function is C(x) = 4x +5(x² -60x + 1044)^1/2. A graph is attached.

e. If x = 14, then the cost of laying the gas line is minimum at $ 156.

Note:

The minimum can be seen in the original Desmos graph. It cannot be seen in an image.