Question

A cable runs along the wall from C to P at a cost of $6 per meter, and straight from P to M at a cost of $10 per meter. If M is 12 meters from the nearest point A on the wall where P lies, and A is 35 meters from C, find the distance from C to P such that the cost of installing the cable is minimized and find this cost.

Answer 1

To minimize the cost of installing the cable, we find the distance from C to P that minimizes the expression 6x + 10(23 - x), where x is the distance from C to P. The distance from C to P that minimizes the cost is 57.5 meters, and the cost is $632.50.

Step-by-step explanation:

To minimize the cost of installing the cable, we need to find the distance from C to P. Let's start by finding the distance from P to M. Since M is 12 meters from the nearest point A and A is 35 meters from C, the distance from P to M is 35 - 12 = 23 meters. Now, the cost of running the cable from C to P is $6 per meter and from P to M is $10 per meter. So, the total cost is 6 * (C to P distance) + 10 * (P to M distance). To minimize the cost, we need to find the value of C to P that minimizes this expression.

Let's denote the distance from C to P as x. The total cost is then 6x + 10(23 - x). Simplifying this expression, we get 6x + 230 - 10x. Combining like terms, we have -4x + 230. To minimize this expression, we take the derivative with respect to x and set it equal to 0.

d/dx(-4x + 230) = -4 = 0

Solving for x, we get x = 57.5. Therefore, the distance from C to P that minimizes the cost is 57.5 meters, and the cost is 6 * 57.5 + 10 * (23 - 57.5) = $632.50.