The variable costs associated with a certain process are $0.65 per item. The fixed costs per day have been calculated as $200 with special costs estimated as $0.02X^2, where X is the size of the production run (that is, number of items produced). Therefore, the total cost function for the production of x items is: TC= 200-0.65x-0.02x^2
a.Calculate the size of the daily run that will minimize cost per item?
b.Find the cost of a days production for a run that minimize cost per item?
To calculate the size of the daily run that will minimize cost per item, you need to find the derivative of the total cost function with respect to x, set it equal to zero, and solve for x.
The total cost function is given as: TC = 200 - 0.65x - 0.02x^2
a. To find the derivative, differentiate the total cost function with respect to x:
d(TC)/dx = -0.65 - 0.04x
b. Set the derivative equal to zero and solve for x:
-0.65 - 0.04x = 0
-0.04x = 0.65
x = 0.65 / -0.04
x ≈ 16.25
So, the size of the daily run that will minimize cost per item is approximately 16.25.
To find the cost of a day's production for the run that minimizes cost per item, substitute the value of x into the total cost function:
TC = 200 - 0.65(16.25) - 0.02(16.25)^2
TC = 200 - 10.56 - 5.29
TC ≈ 184.15
Therefore, the cost of a day's production for a run that minimizes cost per item is approximately $184.15.