Given a total cost function: C= Q3 - 3Q2 + 8Q + 48 ,then find,
TFC, TVC ,
AFC,AVC, AC and MC
the minimum point of MC and the minimum point of AVC
Determine the level of output for which MC=AVC.
To find TFC (total fixed cost), we need to analyze the given cost function. In this case, TFC refers to the fixed costs that do not vary with the level of output. From the given cost function C = Q^3 - 3Q^2 + 8Q + 48, we can deduce that the fixed costs are represented by the constant term, which is 48. Therefore, TFC = 48.
To find TVC (total variable cost), we need to exclude the fixed costs from the total cost function. In this case, TVC refers to the variable costs that do vary with the level of output. From the given cost function C = Q^3 - 3Q^2 + 8Q + 48, we can deduce that TVC is the rest of the cost function excluding the constant term. Therefore, TVC = Q^3 - 3Q^2 + 8Q.
To find AFC (average fixed cost), we divide TFC by the level of output (Q). In this case, AFC = TFC / Q. Since TFC = 48, AFC = 48 / Q.
To find AVC (average variable cost), we divide TVC by the level of output (Q). In this case, AVC = TVC / Q. Since TVC = Q^3 - 3Q^2 + 8Q, AVC = (Q^3 - 3Q^2 + 8Q) / Q = Q^2 - 3Q + 8.
To find AC (average cost), we divide the total cost (C) by the level of output (Q). In this case, AC = C / Q = (Q^3 - 3Q^2 + 8Q + 48) / Q = Q^2 - 3Q + 8 + (48 / Q).
To find MC (marginal cost), we take the derivative of the total cost function with respect to the level of output (Q). In this case, MC = dC / dQ = 3Q^2 - 6Q + 8.
To find the minimum point of MC, we need to set its derivative equal to zero and solve for Q.
3Q^2 - 6Q + 8 = 0
Using the quadratic formula, we can find the roots:
Q = (6 ± sqrt((-6)^2 - 4(3)(8))) / (2(3))
Q = (6 ± sqrt(36 - 96)) / 6
Q = (6 ± sqrt(-60)) / 6
Since the square root of a negative number is not a real number, there are no real solutions for Q in this case. Thus, there is no minimum point of MC.
To find the minimum point of AVC, we need to set its derivative equal to zero and solve for Q.
Q^2 - 3Q + 8 = 0
Using the quadratic formula, we can find the roots:
Q = (3 ± sqrt((-3)^2 - 4(1)(8))) / (2(1))
Q = (3 ± sqrt(9 - 32)) / 2
Q = (3 ± sqrt(-23)) / 2
Since the square root of a negative number is not a real number, there are no real solutions for Q in this case. Thus, there is no minimum point of AVC.
To determine the level of output for which MC = AVC, we need to set the two equations equal to each other and solve for Q.
3Q^2 - 6Q + 8 = Q^2 - 3Q + 8
2Q^2 = 0
Q^2 = 0
Q = 0
Therefore, the level of output for which MC = AVC is 0.
To find the Total Fixed Cost (TFC), we need to determine the cost that does not vary with the level of output. In this case, TFC can be obtained by finding the value of C when Q = 0.
Given: C = Q^3 - 3Q^2 + 8Q + 48
When Q = 0:
C = (0)^3 - 3(0)^2 + 8(0) + 48
C = 48
Therefore, TFC = 48.
To find the Total Variable Cost (TVC), we need to subtract the TFC from the total cost.
TVC = C - TFC
TVC = Q^3 - 3Q^2 + 8Q + 48 - 48
TVC = Q^3 - 3Q^2 + 8Q
Now, let's calculate the Average Fixed Cost (AFC), Average Variable Cost (AVC), Average Cost (AC), and Marginal Cost (MC).
AFC = TFC / Q
AFC = 48 / Q
AVC = TVC / Q
AVC = (Q^3 - 3Q^2 + 8Q) / Q
AVC = Q^2 - 3Q + 8
AC = C / Q
AC = (Q^3 - 3Q^2 + 8Q + 48) / Q
AC = Q^2 - 3Q + 8 + 48/Q
MC = dC / dQ
MC = 3Q^2 - 6Q + 8
To find the minimum point of MC, we need to calculate the derivative of MC and set it equal to zero.
dMC / dQ = 6Q - 6
Setting this equal to zero and solving for Q:
6Q - 6 = 0
6Q = 6
Q = 1
So, the minimum point of MC is when Q = 1.
To find the minimum point of AVC, we need to calculate the derivative of AVC and set it equal to zero.
dAVC / dQ = 2Q - 3
Setting this equal to zero and solving for Q:
2Q - 3 = 0
2Q = 3
Q = 3/2
So, the minimum point of AVC is when Q = 3/2.
To determine the level of output for which MC = AVC, set the equations for MC and AVC equal to each other and solve for Q:
3Q^2 - 6Q + 8 = Q^2 - 3Q + 8
2Q^2 - 3Q = 0
Q(2Q - 3) = 0
This equation has two solutions: Q = 0 and Q = 3/2. However, we need to consider that the level of output cannot be zero (Q = 0) since it results in no production. Therefore, the level of output for which MC = AVC is Q = 3/2.
To find the answers to this question, we can break it down step by step.
Step 1: Find the Total Fixed Cost (TFC)
Total Fixed Cost (TFC) is the portion of the cost that doesn't change with the level of output. In this case, TFC is represented by the constant term in the total cost function. So, TFC = 48.
Step 2: Find the Total Variable Cost (TVC)
Total Variable Cost (TVC) is the portion of the cost that varies with the level of output. In this case, it is represented by the terms in the total cost function that have coefficients multiplied by Q. So, TVC = Q^3 - 3Q^2 + 8Q.
Step 3: Find the Average Fixed Cost (AFC)
Average Fixed Cost (AFC) is calculated by dividing the Total Fixed Cost (TFC) by the level of output (Q). So, AFC = TFC / Q.
Step 4: Find the Average Variable Cost (AVC)
Average Variable Cost (AVC) is calculated by dividing the Total Variable Cost (TVC) by the level of output (Q). So, AVC = TVC / Q.
Step 5: Find the Average Cost (AC)
Average Cost (AC), also known as Average Total Cost (ATC), is calculated by dividing the Total Cost (TC) by the level of output (Q). So, AC = (TFC + TVC) / Q.
Step 6: Find the Marginal Cost (MC)
Marginal Cost (MC) is calculated by taking the derivative of the Total Cost (TC) function with respect to the level of output (Q). So, MC = dC/dQ.
Step 7: Find the minimum point of MC
To find the minimum point of MC, set the derivative equal to zero and solve for Q. Then plug that value of Q back into the MC equation to find the minimum point.
Step 8: Find the minimum point of AVC
Similarly, to find the minimum point of AVC, set the derivative of AVC equal to zero and solve for Q. Then plug that value of Q back into the AVC equation to find the minimum point.
Step 9: Find the level of output for which MC = AVC
Set the MC equation equal to the AVC equation and solve for Q.
By following these steps, you should be able to find the values for TFC, TVC, AFC, AVC, AC, MC, the minimum point of MC and AVC, and the level of output for which MC = AVC.