For additional income, Franklin plans to build craft rocking chairs and sell them at the fair. He has found that people are willing to pay less per rocking chair if he builds more rocking chairs. By building $q$ rocking chairs to sell, he can price each chair at $144-8q$ dollars, which means his revenue will be $q(144-8q)$ dollars. It will cost Franklin $\$16$ in supplies to build each rocking chair, and he just bought a new power drill and saw for $\$312$ . So, his costs will be $16q+312$ dollars. Therefore, if he builds and sells $q$ rocking chairs, his profit in dollars will be $q(144-8q)-(16q+312)$ . This expression can be simplified to $-8q^2+128q-312$ and then written in factored form as $-8(q-3)(q-13)$ . What do the numbers $144-8q$ $0$ and $144-8q$ $1$ represent in the expression?

View step-by-step help

Home

Math Problems

Algebra 2

Interpret parts of quadratic expressions: word problems

Full solution

Q. For additional income, Franklin plans to build craft rocking chairs and sell them at the fair. He has found that people are willing to pay less per rocking chair if he builds more rocking chairs. By building

q

rocking chairs to sell, he can price each chair at

144-8q

dollars, which means his revenue will be

q(144-8q)

dollars. It will cost Franklin

\$16

in supplies to build each rocking chair, and he just bought a new power drill and saw for

\$312

. So, his costs will be

16q+312

dollars. Therefore, if he builds and sells

q

rocking chairs, his profit in dollars will be

q(144-8q)-(16q+312)

. This expression can be simplified to

-8q^2+128q-312

and then written in factored form as

-8(q-3)(q-13)

. What do the numbers

144-8q

0

and

144-8q

1

represent in the expression?

Identify Factored Form: question_prompt: What do the numbers $3$ and $13$ represent in the expression?
Interpret Factors: Step $1$ : Identify the factored form of the profit expression. $\newline$ The profit expression is given as $-8(q-3)(q-13)$ .
Solve for $q$ : Step $2$ : Interpret the factors in the context of the problem. $\newline$ The factors $(q-3)$ and $(q-13)$ in the expression $-8(q-3)(q-13)$ represent the values of $q$ that make the profit zero.
Conclude Meaning of Numbers: Step $3$ : Solve for $q$ in each factor. $\newline$ Setting each factor to zero gives: $\newline$ $q - 3 = 0$ → $q = 3$ $\newline$ $q - 13 = 0$ → $q = 13$
Conclude Meaning of Numbers: Step $3$ : Solve for $q$ in each factor. $\newline$ Setting each factor to zero gives: $\newline$ $q - 3 = 0$ → $q = 3$ $\newline$ $q - 13 = 0$ → $q = 13$ Step $4$ : Conclude what the numbers $3$ and $13$ represent. $\newline$ The numbers $3$ and $13$ represent the number of rocking chairs Franklin needs to build and sell to break even, meaning his total revenue equals his total costs at these points.