Score: $6 / 14 \quad 5 / 10$ answered
Question 7
You decide to begin selling caramel apples at the local park and swap. Your cost for each caramel apple is $\$ 2.15$ plus you have to pay a fixed weekly fee of $\$ 230$ for the booth. Your plan is to sell each caramel apple for $\$ 3.37$.
a. Write a function, $C(n)$, to represent your total costs for the week if you sell $n$ caramel apples.
\[
C(n)=
\]
b. Write a function, $R(n)$, to represent the revenue from the sale of $n$ caramel apples during the week.
\[
R(n)=
\]
c. Write a function, $P(n)$, that represents the profits for selling $n$ caramel apples in a given week.
\[
P(n)=
\]
d. How many items must you sell to break even?
caramel apples

Step 1 ：Write a function, $C(n)$, to represent your total costs for the week if you sell $n$ caramel apples. The cost for each caramel apple is $2.15 and there is a fixed weekly fee of $230. So, the total cost for the week if we sell n caramel apples would be $2.15n + $230. Therefore, $C(n) = 2.15n + 230$.

Step 2 ：Write a function, $R(n)$, to represent the revenue from the sale of $n$ caramel apples during the week. Each caramel apple is sold for $3.37. So, the total revenue from the sale of n caramel apples during the week would be $3.37n. Therefore, $R(n) = 3.37n$.

Step 3 ：Write a function, $P(n)$, that represents the profits for selling $n$ caramel apples in a given week. Profit is revenue minus cost. So, the profit for selling n caramel apples in a given week would be $3.37n - ($2.15n + $230). Therefore, $P(n) = 1.22n - 230$.

Step 4 ：To find the break-even point, set the profit function equal to zero and solve for n. This will give the number of caramel apples we need to sell in order to break even. The break-even point is approximately 189 caramel apples. Therefore, you must sell approximately 189 caramel apples to break even. The final answer is \(\boxed{189}\).