$P(q)=-0.01(q-250)(q-80)$ $\newline$ The given equation gives the profit, $P(q)$ , in dollars, earned by a cupcake bakery when $q$ cupcakes are produced. What is the best interpretation of the number $80$ in this context? $\newline$ Choose $1$ answer: $\newline$ (A) $80$ is a number of cupcakes for which the profit is equal to $\$ 0$ . $\newline$ (B) $80$ is the number of cupcakes that corresponds to the maximum profit. $\newline$ (C) $80$ is the number of cupcakes that corresponds to the minimum profit. $\newline$ (D) $\$ 80$ is the maximum profit, in dollars.

Full solution

P(q)=-0.01(q-250)(q-80)

\newline

The given equation gives the profit,

P(q)

, in dollars, earned by a cupcake bakery when

q

cupcakes are produced. What is the best interpretation of the number

80

in this context?

\newline

Choose

1

answer:

\newline

(A)

80

is a number of cupcakes for which the profit is equal to

\$ 0

\newline

(B)

80

is the number of cupcakes that corresponds to the maximum profit.

\newline

(C)

80

is the number of cupcakes that corresponds to the minimum profit.

\newline

(D)

\$ 80

is the maximum profit, in dollars.

Profit Equation Analysis: The profit equation is $P(q)=-0.01(q-250)(q-80)$ . To understand what the number $80$ represents, we need to look at the factors of the quadratic equation.
Factors of Quadratic Equation: The factors $(q-250)$ and $(q-80)$ indicate the values of $q$ for which the profit $P(q)$ will be zero. This is because if $q$ equals either $250$ or $80$ , one of the factors will be zero, making the entire product zero.
Zero Profit Point: Since we're looking at the number $80$ , when $q=80$ , the factor $(q-80)$ becomes zero, which means the profit $P(q)$ is zero. This indicates that at $80$ cupcakes, the bakery does not make a profit.
Interpretation of Number $80$ : Therefore, the best interpretation of the number $80$ in this context is that it is the number of cupcakes for which the profit is equal to $\$0$ .