Question

The total cost (in dollars) of producing x college textbooks is C(x)=48x+97000.(a) What are the fixed costs?(b) What is the marginal cost per book?(c) What is the total cost of producing 1200 books? 32,000 books?(d) What is the average cost when 1200 books are produced? when 32,000 books are produced?

Answer 1

Given the question in the question tab, the following are the solution steps for answering the question

STEP 1: Write the function for the total cost

`C(x)=48x+97000`

STEP 2: Explain Fixed cost

The term fixed cost refers to a cost that does not change with an increase or decrease in the number of goods or services produced or sold.

`Total\text{ cost}=\text{Variable cost + }Fixed\text{ cost}`

Since Fixed cost does not change from the function of the Total cost, hence the fixed cost is 97000 dollars

STEP 3: Calculate the Marginal cost

Marginal cost is the change in total cost when another unit is produced. Marginal cost is the derivative of the cost function.

By formula;

`\begin{gathered} \text{Marignal Cost}=(dC(x))/(dx) \\ C(x)=48x+97000 \\ \text{Derivative of C(x) gives 48+0 since differentiation of a constant is zero} \\ \text{Marignal Cost}=(dC(x))/(dx)=48 \end{gathered}`

Hence, the marginal cost per book is 48 dollars

STEP 4: Calculate total cost for producing 1200 books

`\begin{gathered} C(x)=48x+97000 \\ x=1200\text{ books} \\ C(x)=48(1200)+97000=57600+97000=154600 \\ \\ \text{When x=32000 books} \\ C(x)=48x+97000 \\ C(x)=48(32000)+97000=1536000+97000=1633000 \end{gathered}`

The total cost of producing 1200 books is 154600 dollars while the total cost of producing 32,000 books is 1633000 dollars.

STEP 5: Calculate the average cost when 1200 books and 32000 books are produced

Average cost (AC), also known as average total cost (ATC), is the average cost per unit of output. To find it, divide the total cost (TC) by the quantity the firm is producing (Q).

`\begin{gathered} \text{Average Cost}=(C(x))/(x) \\ \text{when x = 1200} \\ \text{Average Cost}=(C(1200))/(1200) \\ \text{ Recall from Step 4 that C(1200)=154600} \\ \text{Average Cost for 1200 books}=(C(1200))/(1200)=(154600)/(1200)=128.833333\approx128.83 \\ \\ \text{when x = 32000} \\ \text{Average Cost}=(C(32000))/(1200) \\ \text{Recall from Step 4 that C(32000)=16}33000 \\ \text{Average Cost for 32000 books}=(C(32000))/(32000)=(1633000)/(32000)=51.03125\approx51.03 \end{gathered}`

It can be seen that the average cost for producing 1200 books is approximately 128.83 dollars while the average cost for producing 32000 books is approximately 51.03 dollars