Final answer:
To determine the smallest number of signs that Allen can sell to have an income of $280, we use the income equation I = x(43 - x) and solve for x. By solving the quadratic equation, we find that the smallest number of signs Allen can sell while still achieving $280 in income is 8 signs.
Step-by-step explanation:
The question involves an equation representing the relationship between the price charged for each sign and the number of signs sold by Allen's business. To find the smallest number of signs he can sell while still achieving an income of $280 in one week, we need to set up the equation for his income. His income (I) can be expressed as the product of the price per sign (x) and the number of signs sold (43 - x).
I = x(43 - x)
To achieve an income of $280, we set the income equation equal to 280 and solve for x:
280 = x(43 - x)
This is a quadratic equation which can be solved by expanding and rearranging the terms, then finding the roots of the quadratic equation. Let's find the solutions:
- 0 = x² - 43x + 280
- x² - 43x + 280 = 0
- Factors of 280 that add up to 43 are 35 and 8.
- (x - 35)(x - 8) = 0
- So, x = 35 or x = 8
If x = 35, then 43 - x = 43 - 35 = 8 signs sold.
If x = 8, then 43 - x = 43 - 8 = 35 signs sold.
Therefore, the smallest number of signs that can be sold to have an income of $280 is 8 signs.