The rectangular plot in Figure 23 has size 100 ft by 200 ft. Pipe is to be laid from A to a point P on side BC and from there to C. The cost of laying pipe along the side of the plot is $ 50/ft and the cost through the plot is $ 70/ft (since it is underground).

Question

The rectangular plot in Figure 23 has size 100 ft by 200 ft. Pipe is to be laid from A to a point P on side BC and from there to C. The cost of laying pipe along the side of the plot is $ 50/ft and the cost through the plot is $ 70/ft (since it is underground).

(a) Let f(x) be the total cost, where x is the distance from P to B. Determine f(x), but note that f is discontinuous at x = 0 (when x = 0, the cost of the entire pipe is $ 50/ft). What is the minimal cost?

(b) If the cost along the sides is $ 64/ft, what is the minimal cost?

Answer 1

To solve part (a), we need to determine the function f(x) that represents the total cost of laying the pipe from A to P and from P to C.

First, let's consider the different possibilities for laying the pipe. We have two options: laying the pipe along the sides of the plot or laying it through the plot. The cost of laying the pipe along the sides is $50/ft, and the cost of laying it through the plot is $70/ft.

Let's break down the problem into two cases:

1) When x > 0:
In this case, the pipe will be laid along the sides for a distance of x and then connected to point P. From point P to C, the pipe will be laid through the plot. The total cost can be expressed as:
f(x) = 50x + 70 * (200 - x)

2) When x = 0:
Here, the pipe is laid entirely along the sides from A to C. The total cost is simply the length of the sides multiplied by the cost per foot:
f(x) = 50 * 300

Since the function f(x) is discontinuous at x = 0, we need to find the limit as x approaches 0 from the positive side to determine the minimal cost.

lim(x->0+) f(x) = lim(x->0+) (50x + 70 * (200 - x))
= lim(x->0+) (50x) + lim(x->0+) (70 * (200 - x))
= 0 + 70 * 200
= 14000

So the minimal cost, f(x), is $14,000 when x = 0.

Now let's move on to part (b) of the question.

If the cost along the sides is $64/ft, we need to revise our functions accordingly.

1) For x > 0:
f(x) = 64x + 70 * (200 - x)

2) For x = 0:
f(x) = 64 * 300

Again, we need to find the minimal cost by taking the limit as x approaches 0 from the positive side.

lim(x->0+) f(x) = lim(x->0+) (64x + 70 * (200 - x))
= 0 + 70 * 200
= 14000

So the minimal cost when the cost along the sides is $64/ft is still $14,000.