Answer:
To answer the questions, let's analyze the given cost function:
c(x) = 800 + 70x - 0.3x^2
a. To find the average cost per appliance of producing the first 80 appliances, we divide the total cost of producing 80 appliances by 80:
Average cost = c(80) / 80
Substituting x = 80 into the cost function:
Average cost = (800 + 70(80) - 0.3(80)^2) / 80
Calculating this expression will give us the average cost per appliance.
b. The marginal cost represents the change in cost when one additional unit is produced. To find the marginal cost when 80 appliances are produced, we need to calculate the derivative of the cost function with respect to x:
c'(x) = 70 - 0.6x
Substituting x = 80 into the derivative:
Marginal cost = c'(80) = 70 - 0.6(80)
This will give us the marginal cost at 80 appliances.
c. To show that the marginal cost when 80 appliances are produced is approximately the cost of producing one more appliance after the first 80 have been made, we can calculate the cost of producing 81 appliances and compare it with the marginal cost at 80 appliances.
To calculate the cost of producing 81 appliances directly, we substitute x = 81 into the cost function:
Cost of producing 81 appliances = c(81) = 800 + 70(81) - 0.3(81)^2
By comparing this cost with the marginal cost at 80 appliances, we can observe if they are approximately equal.
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Explanation: