A cable runs along the wall from C to P at a cost of $3 per meter, and straight from P to M at a cost of $5 per meter. If M is 16 meters from the nearest point A on the wall where P lies, and A is 50 mters from C, find the distance from C to P such that the cost of installing the cable is minimized and find this cost
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Let x be the distance from C to P.
The cost of installing the cable is 3x + 5(16 - x) = 3x + 80 - 5x = 80 + -2x
To minimize the cost, we need to find the value of x that minimizes the cost.
Taking the derivative of the cost function with respect to x, we get -2. Setting this equal to 0 and solving for x, we get x = 40.
Therefore, the distance from C to P is 40 meters and the cost of installing the cable is minimized to $80.
To find the distance from C to P that minimizes the cost of installing the cable, we need to determine the total cost function and find its minimum value.
Let's label the distance from C to P as x. Then, the distance from P to M is 16 - x (since M is 16 meters from the nearest point A).
The cost of running the cable from C to P is $3 per meter, so the cost of this segment is 3x.
The cost of running the cable from P to M is $5 per meter, so the cost of this segment is 5(16 - x).
The total cost function, C(x), is the sum of these costs:
C(x) = 3x + 5(16 - x) = 3x + 80 - 5x = -2x + 80
To find the value of x that minimizes the cost, we can take the derivative of C(x) with respect to x and set it equal to zero:
C'(x) = -2
Setting -2x + 80 equal to zero, we can solve for x:
-2x + 80 = 0
2x = 80
x = 40
Therefore, the distance from C to P that minimizes the cost is 40 meters.
To find the minimum cost, we substitute this value of x back into the total cost function:
C(40) = -2(40) + 80 = 80 - 80 = 0
Therefore, the cost of installing the cable with this distance is $0.
To find the distance from C to P that minimizes the cost of installing the cable, we first need to determine the total cost function.
Let's break down the cost. The cable runs from C to P at a cost of $3 per meter and then straight from P to M at a cost of $5 per meter.
The cost of the cable from C to P is given by: cost_CP = distance_CP * $3.
The cost of the cable from P to M is given by: cost_PM = distance_PM * $5.
Since M is 16 meters from the nearest point A on the wall where P lies, we can derive that distance_PM = distance_CP - 16.
Therefore, the total cost function can be expressed as follows:
Total cost = cost_CP + cost_PM = distance_CP * $3 + (distance_CP - 16) * $5.
Now, to minimize the cost, we can differentiate the total cost function with respect to distance_CP, set the derivative equal to zero, and solve for distance_CP.
d(total cost)/d(distance_CP) = 3 + 5 - 5 = 0 (since the derivative of $3 is 0).
8 = 5(distance_CP - 16).
8/5 = distance_CP - 16.
distance_CP = 8/5 + 16 = 8/5 + 80/5 = 88/5 = 17.6 meters.
Therefore, the distance from C to P that minimizes the cost is 17.6 meters.
To calculate the minimum cost, substitute the value of distance_CP into the total cost function:
Total cost = distance_CP * $3 + (distance_CP - 16) * $5
Total cost = 17.6 * $3 + (17.6 - 16) * $5
Total cost = $52.8 + 1.6 * $5
Total cost = $52.8 + $8
Total cost = $60.8
Thus, the cost of installing the cable when the distance from C to P is minimized is $60.8.