16. Mrs. Hunneycutt makes dog sweaters. Her profit each month is a function of the price she charges for each sweater. Her total profits, y (in hundreds of dollars), based
on the sales price, x, can be modeled by the function y=-3x2 + 39x - 108.
Identify any zeros of the function, and interpret what the zeros mean in terms of the
situation

The total profits will be zero at sales prices of $4 and $9

Step-by-step explanation:

∵ The function is y = -3x² + 39x - 108, where

The zeroes of the function are the values of x at y = 0

→ To find the zeroes of the function above equate y by 0

∵ 0 = -3x² + 39x - 108

→ Switch the two sides

∴ -3x² + 39x - 108 = 0

→ Divide all terms by -3 to simplify the equation

∴ x² - 13x + 36 = 0

→ Let us factor the trinominal into two brackets

∵ The last term is +ve and the middle term is -ve

∴ The two brackets have (-) sign in the middle ⇒ (... - ...)(... - ...)

∵ x² = (x)(x)

∵ 36 = (4)(9)

∵ (4)(x) + (9)(x) = 4x + 9x = 13x ⇒ the value of the middel term

→ That means the factors of the trinomial are (x - 4) and (x - 9)

∴ x² - 13x + 36 = (x - 4)(x - 9)

→ Replace the trinomial by its factors

∴ (x - 4)(x - 9) = 0

→ Equate each bracket by 0 to find x

∵ x - 4 = 0

→ Add 4 to both sides

∴ x = 4

∵ x - 9 = 0

→ Add 4 to both sides

∴ x = 9

∴ The zeroes of the functions are 4 and 9

∵ x represents the sales price in dollars

∴ The sales prices which make zero profit are $4 and $9

∴ The total profits will be zero at sales prices of $4 and $9