A jewelry artist is selling necklaces at an art fair. It costs
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. a) Will the artist make a profit if she sells 15 necklaces?
The Correct Answer and Explanation is:
To determine whether the artist makes a profit by selling 15 necklaces, we will use the inequality:12n>135+4.50n12n > 135 + 4.50n12n>135+4.50n
Here:
- nnn is the number of necklaces sold
- 12n12n12n represents the revenue from selling nnn necklaces at $12 each
- 135135135 is a fixed cost (for example, the booth fee or materials)
- 4.50n4.50n4.50n represents the cost per necklace
Step 1: Plug in n=15n = 15n=15
12(15)>135+4.50(15)12(15) > 135 + 4.50(15)12(15)>135+4.50(15)180>135+67.5180 > 135 + 67.5180>135+67.5180>202.5180 > 202.5180>202.5
This statement is not true. 180 is less than 202.5.
Final Answer:
No, the artist will not make a profit if she sells 15 necklaces.
Explanation:
This problem models a real-world business scenario using an inequality. The goal is to determine whether the artist earns more money from sales than she spends on costs. The total revenue comes from selling necklaces at $12 each, which is expressed as 12n12n12n. The total cost is the sum of a fixed cost ($135) and a variable cost of $4.50 per necklace, written as 135+4.50n135 + 4.50n135+4.50n.
The inequality 12n>135+4.50n12n > 135 + 4.50n12n>135+4.50n sets up the condition for profit. Profit occurs when income exceeds expenses. To check if selling 15 necklaces leads to a profit, we substitute n=15n = 15n=15 into both sides of the inequality.
The left-hand side becomes 12×15=18012 \times 15 = 18012×15=180, and the right-hand side becomes 135+4.50×15=135+67.5=202.5135 + 4.50 \times 15 = 135 + 67.5 = 202.5135+4.50×15=135+67.5=202.5. The comparison is 180>202.5180 > 202.5180>202.5, which is false. Since the revenue is not greater than the total cost, no profit is made.
This illustrates how important it is to consider both fixed and variable costs when setting prices. Even though $12 seems like a high selling price, the combined costs make it hard to earn a profit with just 15 sales. The artist would need to sell more necklaces to make the business viable.
