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 – 8q2+128q–312 and then written in factored form as – 8(q–3)(q–13). What do the numbers 3 and 13 represent in the expression?

Question

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  – 8q2+128q–312 and then written in factored form as  – 8(q–3)(q–13). What do the numbers 3 and 13 represent in the expression?

Bytelearn · Accepted Answer

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. 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. 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. Setting each factor to zero gives: q - 3 = 0  →  q = 3 q - 13 = 0  →  q = 13  Step 4: Conclude what the numbers 3 and 13 represent. 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.

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