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. Let x be the distance from C to P. If M is 8 meters from the nearest point A on the wall where P lies, and A is 18 meters from point C, find x such that the cost of installing the cable is minimized and find this cost.
Ok, I am lost without a picture.
To find the distance x from C to P that minimizes the cost of installing the cable, we need to consider the different costs for running the cable along the wall and straight to M.
Let's break down the problem and identify the distances involved:
1. Distance from C to A: 18 meters
2. Distance from A to M: 8 meters
3. Distance from C to P: x meters
4. Distance from P to M: (x + 8) meters (since M is 8 meters from A)
Now, let's calculate the cost of installing the cable:
1. Cost for running the cable from C to A: $6 x 18 = $108 (since it costs $6 per meter)
2. Cost for running the cable from P to M: $10 x (x + 8) (since it costs $10 per meter)
Therefore, the total cost, denoted as C(x), is given by:
C(x) = $108 + $10(x + 8)
C(x) = $108 + $10x + $80
C(x) = $10x + $188
To minimize the cost C(x), we can differentiate it with respect to x and set the derivative equal to zero:
dC(x)/dx = 10
Setting the derivative equal to zero gives us:
10 = 0
This means that the cost C(x) is a linear function, and its minimum value occurs at the endpoint of the given interval.
Since x represents the distance from C to P, the minimum cost C(x) occurs when x = 0 or x = 18. However, x cannot be 0 since that would mean no cable is installed.
Therefore, the distance x from C to P that minimizes the cost of installing the cable is x = 18 meters, and the corresponding cost is C(18) = $10(18) + $188 = $368.
Therefore, the minimum cost of installing the cable is $368.
To find the distance x from C to P that minimizes the cost of installing the cable, we need to create a function for the cost.
Let's break down the total cost based on the given information:
1. Cost of running the cable from C to P: $6 per meter, which is equal to 6x.
2. Cost of running the cable from P to M: $10 per meter for a straight line.
Since M is 8 meters away from the nearest point A on the wall where P lies, and A is 18 meters from point C, the distance from P to M is (18 - x - 8), which simplifies to (10 - x).
Therefore, the total cost function, f(x), for installing the cable is:
f(x) = 6x + 10(10 - x)
To find the value of x that minimizes the cost, we can take the derivative of f(x) with respect to x and set it equal to zero:
f'(x) = 6 - 10 = 0
Solving for x:
6 - 10 = 0
-4 = 0
Since -4 does not equal zero, there is no solution to f'(x) = 0.
Therefore, we need to analyze the endpoints of the given interval to find the minimum cost.
Given that A is 18 meters from C and M is 8 meters from A, the distance from C to M is equal to (18 + 8) = 26 meters.
Plugging in the values of x = 0 (C) and x = 26 (M) into the total cost function:
f(0) = 6(0) + 10(10 - 0) = 0 + 100 = $100
f(26) = 6(26) + 10(10 - 26) = 156 + (-160) = $-4
Since the distance between C and P cannot be negative, the minimum cost occurs at the endpoint x = 0 (C).
Therefore, the distance x from C to P that minimizes the cost of installing the cable is 0, and the cost is $100.