q 0 125 250 375 500 625 750
C'(q) 26 24 30 26 23 25 29
a) If the fixed cost is $8500, use the average of left- and right-hand sums to determine the total cost of producing 500 units.
Answer: $
b) How much would the total cost increase if production were increased one unit, to 501 units?
Answer: $
To find the answer to these questions, we can use the information given about the production quantity and cost function C'(q). The derivative C'(q) represents the rate at which the cost function C(q) is changing with respect to the quantity produced, i.e., the cost per unit.
a) To determine the total cost of producing 500 units using the average of left and right-hand sums, first, let's calculate the left-hand sum and right-hand sum for the given data:
Left-hand sum:
Add up the products of the difference in q (change in q) and the corresponding C'(q) values for all values of q less than or equal to 500.
Left-hand sum = (q2 - q1) * C'(q1) + (q3 - q2) * C'(q2) + ... + (500 - q6) * C'(q6)
Right-hand sum:
Add up the products of the difference in q (change in q) and the corresponding C'(q) values for all values of q greater than or equal to 500.
Right-hand sum = (q1 - q0) * C'(q0) + (q2 - q1) * C'(q1) + ... + (q7 - 500) * C'(q7)
Average of left and right-hand sums:
Calculate the average of the left-hand sum and the right-hand sum.
Total cost of producing 500 units = (Left-hand sum + Right-hand sum) / 2
Substituting the values from the given data:
Left-hand sum = (500 - 375) * 26 + (375 - 250) * 30 + (250 - 125) * 24 = 11,500
Right-hand sum = (500 - 375) * 23 + (625 - 500) * 25 + (750 - 625) * 29 = 12,000
Total cost of producing 500 units = (11,500 + 12,000) / 2 = 11,750
Therefore, the total cost of producing 500 units is $11,750 if the fixed cost is $8500.
b) To calculate how much the total cost would increase if production were increased by one unit, we need to find the differential cost, which is the change in cost when production increases by one unit.
Differential cost = C'(q) * Δq
where C'(q) is the rate of change of the cost function with respect to the quantity produced (given in the problem), and Δq is the change in quantity.
By substituting the given C'(q) value for the current production level into the formula:
Differential cost = C'(500) * 1 = 23
Therefore, if production were increased one unit to 501 units, the total cost would increase by $23.