To determine the break-even point for the bookstore, we need to find the number of books, [tex]\( n \)[/tex], that must be sold so that the revenue equals the total costs. The steps to solve this are as follows:
1. Define the costs and prices:
- Fixed cost per day: [tex]\( \$105 \)[/tex]
- Variable cost per book: [tex]\( \$12 \)[/tex]
- Selling price per book: [tex]\( \$15 \)[/tex]
2. Set up the break-even equation:
- The revenue from selling [tex]\( n \)[/tex] books is given by:
[tex]\[
\text{Revenue} = n \times \text{selling price per book} = n \times \$15
\][/tex]
- The total cost is the sum of the fixed cost and the variable cost (cost per book):
[tex]\[
\text{Total Cost} = \text{Fixed Cost} + n \times \text{Variable Cost per Book} = \$105 + n \times \$12
\][/tex]
3. Equate the revenue to the total cost to find the break-even point:
[tex]\[
n \times \$15 = \$105 + n \times \$12
\][/tex]
4. Solve for [tex]\( n \)[/tex]:
[tex]\[
15n = 105 + 12n
\][/tex]
[tex]\[
15n - 12n = 105
\][/tex]
[tex]\[
3n = 105
\][/tex]
[tex]\[
n = \frac{105}{3}
\][/tex]
[tex]\[
n = 35
\][/tex]
Therefore, the number of books that must be sold to break even is [tex]\( n = 35 \)[/tex].
So, the correct answer is:
[tex]\[ \boxed{35} \][/tex]
