Question

Suppose a data provider wants to run a new fiber optic cable from terminal box A, located on the shore of a large lake, to terminal box B, on an island in the lake. It costs 800 per mile to run cable across land and1000 per mile to run it underwater. Determine the number of miles of cable that should be run across the land and the number of miles of cable that should be run underwater to minimize the cost to the company. Your answers should be accurate to the nearest half mile.

Answer 1

To minimize the cost of running the fiber optic cable, we need to determine the number of miles of cable that should be run across the land and the number of miles of cable that should be run underwater. The number of miles of cable that should be run across the land is approximately 3/2 times the total distance, and the number of miles of cable that should be run underwater is approximately 1/2 times the total distance.

Step-by-step explanation:

To minimize the cost of running the fiber optic cable, we need to determine the number of miles of cable that should be run across the land and the number of miles of cable that should be run underwater. Let's assume the distance from terminal box A to terminal box B is d miles. We'll let x represent the number of miles of cable run across the land and y represent the number of miles of cable run underwater.

Since the cost per mile to run the cable across land is $800 and the cost per mile to run it underwater is $1000, the total cost C can be expressed as C = 800x + 1000y.

However, we also know that the total distance is d miles, so we have the constraint x + y = d.

To minimize the cost, we can use the method of substitution to solve for one variable in terms of the other, and then substitute that expression into the cost equation. Let's solve for y:

y = d - x

Now we can substitute this expression for y into the cost equation:

C = 800x + 1000(d - x) = 800x + 1000d - 1000x = (1000 - 800)x + 1000d = 200x + 1000d

Since we want to minimize the cost, we want to find the value of x that minimizes the cost equation. We can do this by finding the vertex of the cost equation, which corresponds to the minimum value. The x-coordinate of the vertex is given by x = -b/2a, where a = 200 and b = 1000d.

So, x = -1000d/2(200) = -500d/200 = -2.5d

Since we can't have a negative number of miles, we can round up to the nearest half mile: x = 3d/2

Substituting this value back into the constraint equation x + y = d, we have (3d/2) + y = d. Solving for y, we get y = d/2.

Therefore, to minimize the cost, the number of miles of cable that should be run across the land is approximately 3/2 times the total distance, and the number of miles of cable that should be run underwater is approximately 1/2 times the total distance.