To determine whether the point [tex]\( n = 150 \)[/tex] is in the loss section, profit section, or break-even section for a new bookstore, we need to evaluate both the cost and revenue functions at [tex]\( n = 150 \)[/tex].
1. The cost function is given by:
[tex]\[
C = 20n + 500
\][/tex]
Substituting [tex]\( n = 150 \)[/tex]:
[tex]\[
C = 20 \times 150 + 500 = 3000 + 500 = 3500
\][/tex]
2. The revenue function is given by:
[tex]\[
r = 25n
\][/tex]
Substituting [tex]\( n = 150 \)[/tex]:
[tex]\[
r = 25 \times 150 = 3750
\][/tex]
3. Now we compare the cost and revenue at [tex]\( n = 150 \)[/tex]:
- Cost at [tex]\( n = 150 \)[/tex] is 3500.
- Revenue at [tex]\( n = 150 \)[/tex] is 3750.
Since the revenue (3750) is greater than the cost (3500), the bookstore is making a profit at [tex]\( n = 150 \)[/tex].
Thus, the point [tex]\( n = 150 \)[/tex] would be in the:
D. Profit section
