suppose that the dollar cost of producing x appliances is c(x)=900 + 130x - 0.4x^2. a. find the average cost…

suppose that the dollar cost of producing x appliances is c(x)=900 + 130x - 0.4x^2. a. find the average cost per appliance of producing the first 150 appliances. b. find the marginal cost when 150 appliances are produced. c. show that the marginal cost when 150 appliances are produced is approximately the cost of producing one more appliance after the first 150 have been made, by calculating the latter cost. (round to the nearest cent as needed.) the average cost per appliance of producing the first 150 appliances is $__/appliance.

suppose that the dollar cost of producing x appliances is c(x)=900 + 130x - 0.4x^2. a. find the average cost per appliance of producing the first 150 appliances. b. find the marginal cost when 150 appliances are produced. c. show that the marginal cost when 150 appliances are produced is approximately the cost of producing one more appliance after the first 150 have been made, by calculating the latter cost. (round to the nearest cent as needed.) the average cost per appliance of producing the first 150 appliances is $__/appliance.

Answer

Explanation:

Step1: Find the total cost of producing 150 appliances

Given the cost - function $c(x)=900 + 130x-0.4x^{2}$, substitute $x = 150$ into the function. $c(150)=900+130\times150 - 0.4\times(150)^{2}$ $=900 + 19500-0.4\times22500$ $=900 + 19500 - 9000$ $=11400$

Step2: Calculate the average cost per appliance

The average cost per appliance $\bar{c}$ when $x = 150$ is $\bar{c}=\frac{c(150)}{150}$. $\bar{c}=\frac{11400}{150}=76$

Step3: Find the derivative of the cost - function

The derivative of $c(x)=900 + 130x-0.4x^{2}$ using the power rule $\frac{d}{dx}(x^{n})=nx^{n - 1}$ is $c^\prime(x)=130-0.8x$.

Step4: Calculate the marginal cost when $x = 150$

Substitute $x = 150$ into $c^\prime(x)$. $c^\prime(150)=130-0.8\times150$ $=130 - 120$ $=10$

Step5: Interpret the marginal cost

The marginal cost when 150 appliances are produced is the cost of producing one more appliance after 150 have been made.

Answer:

a. The average cost per appliance of producing the first 150 appliances is $$76.00$ per appliance. b. The marginal cost when 150 appliances are produced is $$10.00$. c. The marginal cost $c^\prime(x)=130 - 0.8x$. When $x = 150$, $c^\prime(150)=130-0.8\times150=10$. This value represents the approximate cost of producing one more appliance after 150 have been made.