A) To find the exact cost of producing the 41st food processor, we can plug in x=41 into the cost function:
C(41) = 2300 + 20 * 41 - 0.1 * 41^2
C(41) = 2300 + 820 - 16.8
C(41) = 3119.2
So, the exact cost of producing the 41st food processor is $3119.2.
B) To use the marginal cost to approximate the cost of producing the 41st food processor, we can use the derivative of the cost function, which represents the rate of change of the cost with respect to the number of food processors produced.
The derivative of the cost function is:
C'(x) = 20 - 0.2x
The marginal cost at x=41 can be calculated as:
C'(41) = 20 - 0.2 * 41
C'(41) = 20 - 8.2
C'(41) = 11.8
So, the marginal cost at x=41 is $11.8, which represents the average increase in cost per unit increase in the number of food processors produced.
We can use this value to estimate the cost of producing the 41st food processor by adding the marginal cost to the cost of producing the 40th food processor:
C(40) + C'(41) = C(41)
C(40) + 11.8 = 3119.2
We can calculate the cost of producing the 40th food processor using the cost function:
C(40) = 2300 + 20 * 40 - 0.1 * 40^2
C(40) = 2300 + 800 - 16
C(40) = 3084
So, we have:
3084 + 11.8 = 3119.2
This is an approximation of the cost of producing the 41st food processor, which is very close to the exact cost found in (A).