Can someone help with the following: I know how to get the MC and AC; however I'm not sure on the algabraic part.
Given the following short-run total cost function, answer the following questions.
TC = 3q2 + 5q + 300
a. Find the marginal and average cost functions.
b. Show that AC is at its minimum when q = 10 and that MC = AC at this output.
1. Which of the following is NOT a relevant factor when determining incremental cash flows for a new product?
a. The use of high quality factory floor space that is currently unused and therefore could be used to produce the proposed new product.
b. Revenues from an existing product that would be lost as a result of customers switching to the new product.
c. Shipping and installation costs associated with preparing a machine which would be used to produce the new product.
d. The cost of a marketing study that was completed last year related to the new product. This research led to the tentative decision to go ahead with the new product, and the cost of the research was expensed for tax purposes last year.
e. The land which would be used for the new project could be sold to another firm.
MC= 8
Av cost= (3q2+5q+300)/2
To find the marginal cost (MC) and average cost (AC) functions, you need to take the derivative of the total cost (TC) function with respect to quantity (q). Let's break it down step by step:
a. Finding the marginal cost function (MC):
To find the marginal cost, you need to take the derivative of the total cost function with respect to quantity (q).
TC = 3q^2 + 5q + 300
To find MC, differentiate TC with respect to q:
MC = dTC/dq
So, take the derivative of each term separately:
d/dq (3q^2) = 6q
d/dq (5q) = 5
d/dq (300) = 0 (as it's a constant)
Putting it all together, the marginal cost function (MC) will be:
MC = 6q + 5
b. Finding the average cost function (AC):
The average cost is the total cost divided by the quantity:
AC = TC / q
Substituting the total cost function:
AC = (3q^2 + 5q + 300) / q
Simplify the expression:
AC = (3q^2 / q) + (5q / q) + (300 / q)
AC = 3q + 5 + (300 / q)
To find the minimum average cost (AC), you need to differentiate the AC function with respect to q and set it equal to zero. However, we can already see that the AC function is not easily differentiable due to the presence of the term 300/q. Therefore, you have to complete the algebraic steps to find the minimum.
For part b, substitute q = 10 into the AC function:
AC = (3 * 10) + 5 + (300 / 10)
AC = 30 + 5 + 30
AC = 65
Now, you need to show that MC = AC at q = 10. Substitute q = 10 into the MC function:
MC = 6 * 10 + 5
MC = 60 + 5
MC = 65
Since MC = AC = 65 at q = 10, it shows that the average cost (AC) is at its minimum at q = 10, and that MC equals AC at this output.