To determine whether the point [tex]\( n = 100 \)[/tex] falls into the loss, profit, or break-even section, we need to evaluate both the cost function and the revenue function at [tex]\( n = 100 \)[/tex] and then compare the values.
1. Cost Function Evaluation:
The cost function [tex]\( C \)[/tex] is given by:
[tex]\[
C = 20n + 500
\][/tex]
Substituting [tex]\( n = 100 \)[/tex] into the cost function:
[tex]\[
C = 20 \times 100 + 500 = 2000 + 500 = 2500
\][/tex]
2. Revenue Function Evaluation:
The revenue function [tex]\( r \)[/tex] is given by:
[tex]\[
r = 25n
\][/tex]
Substituting [tex]\( n = 100 \)[/tex] into the revenue function:
[tex]\[
r = 25 \times 100 = 2500
\][/tex]
3. Comparison:
Now, we compare the cost [tex]\( C \)[/tex] and the revenue [tex]\( r \)[/tex]:
[tex]\[
C = 2500, \quad r = 2500
\][/tex]
Since [tex]\( C = r \)[/tex], the bookstore is neither making a profit nor incurring a loss; they are at the break-even point.
Therefore, at [tex]\( n = 100 \)[/tex], the point would be in the:
A. Break-even section
