Let's break down the given cost function [tex]\(C(q) = 100q - 2q^2 + 0.5q^3 + 45\)[/tex] step by step.
### Part A: Total Fixed Cost (TFC) and Total Variable Cost (TVC)
Total Fixed Cost (TFC):
The Total Fixed Cost is the portion of the cost that does not change with the output quantity [tex]\(q\)[/tex]. From the given cost function, the constant term "45" represents the Total Fixed Cost.
[tex]\[ \text{TFC} = 45 \][/tex]
Total Variable Cost (TVC):
The Total Variable Cost is the portion of the cost that changes with the output quantity [tex]\(q\)[/tex]. We subtract the TFC from the total cost function to derive the TVC.
[tex]\[ \text{TVC} = 100q - 2q^2 + 0.5q^3 \][/tex]
### Part B: Average Fixed Cost (AFC) and Average Variable Cost (AVC)
Average Fixed Cost (AFC):
The Average Fixed Cost is the Total Fixed Cost divided by the output quantity [tex]\(q\)[/tex].
[tex]\[ \text{AFC} = \frac{TFC}{q} = \frac{45}{q} \][/tex]
Average Variable Cost (AVC):
The Average Variable Cost is the Total Variable Cost divided by the output quantity [tex]\(q\)[/tex].
[tex]\[ \text{AVC} = \frac{TVC}{q} = \frac{100q - 2q^2 + 0.5q^3}{q} = 100 - 2q + 0.5q^2 \][/tex]
### Part C: Average Cost (AC) and Marginal Cost (MC)
Average Cost (AC):
The Average Cost is the total cost (which is the sum of TFC and TVC) divided by the output quantity [tex]\(q\)[/tex].
[tex]\[ \text{AC} = \frac{C(q)}{q} = \frac{100q - 2q^2 + 0.5q^3 + 45}{q} = 100 - 2q + 0.5q^2 + \frac{45}{q} \][/tex]
Marginal Cost (MC):
The Marginal Cost is the derivative of the total cost function with respect to quantity [tex]\(q\)[/tex].
[tex]\[ \text{MC} = \frac{dC(q)}{dq} = 100 - 4q + 1.5q^2 \][/tex]
### Part D: Graphical Representation
To draw the Total Fixed Cost (TFC), Total Variable Cost (TVC), and Total Cost (TC) curves:
1. TFC Curve: This will be a horizontal line at the value 45 on the cost-axis, as TFC is constant regardless of output quantity [tex]\( q \)[/tex].
2. TVC Curve: This curve represents the function [tex]\( 100q - 2q^2 + 0.5q^3 \)[/tex]. It starts from the origin and will increase as [tex]\( q \)[/tex] increases.
3. TC Curve: This is the sum of TFC and TVC. Thus,
[tex]\[ \text{TC} = \text{TFC} + \text{TVC} = 45 + 100q - 2q^2 + 0.5q^3 \][/tex]
Therefore, the TC curve is another cubic function that starts at 45 when [tex]\( q = 0 \)[/tex] and increases as [tex]\( q \)[/tex] increases.
To visualize these functions, you would plot them on a graph with quantity [tex]\( q \)[/tex] on the x-axis and cost on the y-axis.
- TFC will be a horizontal line.
- TVC and TC will start from the origin and 45 respectively and exhibit cubic growth patterns.
I hope this comprehensive step-by-step breakdown clarifies each part of the question!
