suppose that the dollar cost of producing x appliances is c(x)=1000 + 70x - 0.1x^2. a. find the average cost…

suppose that the dollar cost of producing x appliances is c(x)=1000 + 70x - 0.1x^2. a. find the average cost per appliance of producing the first 90 appliances. b. find the marginal cost when 90 appliances are produced. c. show that the marginal cost when 90 appliances are produced is approximately the cost of producing one more. the average cost per appliance of producing the first 90 appliances is $72.11 /appliance. (round to the nearest cent as needed.) the marginal cost when 90 appliances are produced is $52. (round to the nearest cent as needed.) the cost of producing one more appliance beyond 90 appliances is $52. (round to the nearest cent as needed.)

suppose that the dollar cost of producing x appliances is c(x)=1000 + 70x - 0.1x^2. a. find the average cost per appliance of producing the first 90 appliances. b. find the marginal cost when 90 appliances are produced. c. show that the marginal cost when 90 appliances are produced is approximately the cost of producing one more. the average cost per appliance of producing the first 90 appliances is $72.11 /appliance. (round to the nearest cent as needed.) the marginal cost when 90 appliances are produced is $52. (round to the nearest cent as needed.) the cost of producing one more appliance beyond 90 appliances is $52. (round to the nearest cent as needed.)

Answer

Explanation:

Step1: Calculate total cost for 90 appliances

Substitute $x = 90$ into $c(x)=1000 + 70x-0.1x^{2}$. $c(90)=1000 + 70\times90-0.1\times90^{2}=1000 + 6300-0.1\times8100=1000 + 6300 - 810=6490$.

Step2: Calculate average cost per appliance

The average cost $\bar{c}=\frac{c(90)}{90}=\frac{6490}{90}\approx72.11$.

Step3: Find the derivative of cost function

The derivative of $c(x)=1000 + 70x-0.1x^{2}$ is $c^\prime(x)=70 - 0.2x$.

Step4: Calculate marginal cost at $x = 90$

Substitute $x = 90$ into $c^\prime(x)$. $c^\prime(90)=70-0.2\times90=70 - 18=52$.

Step5: Interpret marginal cost

The marginal cost $c^\prime(90)$ represents the approximate cost of producing one - more appliance when 90 appliances have been produced. And we found $c^\prime(90) = 52$, which is the cost of producing one more appliance beyond 90 appliances.

Answer:

a. $$72.11$ b. $$52$ c. The marginal cost $c^\prime(90)=52$ and the cost of producing one more appliance beyond 90 is also $$52$, so the marginal cost when 90 appliances are produced is approximately the cost of producing one more appliance.