Question

The Student Council is hosting an art exhibit for young artists. Each artist must pay $33 for a display booth. Plus

they must donate 5% of their total sales to the school. It costs the artists $6 in materials to make a painting that
each sell for $15. Write and solve an equation to find how many paintings the artists must sell in order to break
even.

Answer 1

`\Large \textsf{Read below}`

Explanation:

`\Large \text{$ \sf f(x) = - 33 + 15x - 6x - (x)/(20)$}`

`\Large \text{$ \sf f(x) = - 33 + (300x)/(20) - (120x)/(20) - (x)/(20)$}`

`\Large \boxed{\text{$ \sf f(x) = - 33 + (179x)/(20)$}}`

`\Large \text{$ \sf 0 = - 33 + (179x)/(20)$}`

`\Large \text{$ \sf (179x)/(20) = 33$}`

`\Large \text{$ \sf 179x = 660$}`

`\Large \text{$ \sf x = (660)/(179)$}`

`\Large \boxed{\boxed{\text{$ \sf x \approx 4$}}}`

Answer 2

`\sf Equation: 14.25x = 6x + 33`

The artists must sell 4 paintings to break even.

Explanation:

Let
`\sf x` be the number of paintings the artists must sell to break even.

Cost:

• The materials cost for each painting is 6, so
  `\sf x` paintings will cost
  `\sf 6x` dollars.
• Each artist must pay a 33 both fee.
• The total cost is
  `\sf 6x + 33` dollars.

Income:

• Each painting is sold for 15, generating
  `\sf 15x` dollars in total revenue.
• However, the artist must donate 5% of their total sales to the school, which amounts to
  `\sf 0.05 * 15x = 0.75x` dollars.
• The artist's net income is
  `\sf 15x - 0.75x = 14.25x` dollars.

Break-even point:

The artist breaks even when their income equals their cost, so:

`\sf Equation: 14.25x = 6x + 33`

Solving the equation:

`\sf 8.25x = 33`

`\sf x = (33)/(8.25)`

`\sf x = 4`

Therefore, the artists must sell 4 paintings to break even.