A jewelry artist is selling necklaces at an art fair. It costs $135 to rent a booth at the fair. The cost of materials for each necklace is $4.50. The artist is selling the necklaces at $12 each. The inequality 12n>135+4.50n represents the situation in which the artist makes a profit. Question Will the artist make a profit if he sells 15 necklaces? Be prepared to explain your reasoning in the next question.
The Correct Answer and Explanation is :
To determine whether the artist will make a profit if they sell 15 necklaces, we need to evaluate the inequality ( 12n > 135 + 4.50n ) for ( n = 15 ).
Step 1: Plug ( n = 15 ) into the inequality.
[
12(15) > 135 + 4.50(15)
]
Step 2: Simplify both sides.
- Left-hand side:
[
12(15) = 180
] - Right-hand side:
[
135 + 4.50(15) = 135 + 67.5 = 202.5
]
Step 3: Compare the two sides.
[
180 \not> 202.5
]
Since ( 180 ) is not greater than ( 202.5 ), the inequality is not satisfied. Therefore, the artist will not make a profit if they sell 15 necklaces.
Explanation (300 words):
The inequality ( 12n > 135 + 4.50n ) describes the condition under which the artist will make a profit. Here, ( n ) represents the number of necklaces sold. The left-hand side, ( 12n ), represents the revenue generated from selling ( n ) necklaces at $12 each. The right-hand side, ( 135 + 4.50n ), represents the total cost, which includes the fixed booth rental fee of $135 and the variable cost of $4.50 per necklace.
Profit is achieved when revenue exceeds total costs, which is why the inequality compares these two quantities. When we substitute ( n = 15 ), the revenue is ( 12(15) = 180 ), while the total cost is ( 135 + 4.50(15) = 202.5 ). Since the revenue ($180) is less than the total cost ($202.5), the artist does not make a profit.
This result suggests the artist must sell more necklaces to cover costs and make a profit. To find the exact number of necklaces required for profitability, solve the inequality ( 12n > 135 + 4.50n ). Simplifying gives:
[
12n – 4.50n > 135 \quad \implies \quad 7.50n > 135 \quad \implies \quad n > 18.
]
Thus, the artist needs to sell more than 18 necklaces to make a profit. Since selling only 15 necklaces doesn’t meet this threshold, the artist incurs a loss.