Question

using your math

1. the area of a rectangle is given by the expression $10n + 35$, in square feet. its width is 5 feet as shown. its length is an unknown expression in terms of the variable $n$.

(a) write the area of the rectangle as the product of 5 and another binomial.
(b) explain why the binomial you wrote in (a) must be the length of the rectangle.
(c) test to see if $10n + 35$ and your answer in (a) are equivalent by substituting $n=2$ into both. show your substitution and calculations.
$10n + 35$:
expression from (a):

Explanation:

Step1: Factor the area expression

Factor out 5 from $10n+35$:
$10n+35 = 5(2n+7)$

Step2: Explain rectangle area relationship

Area of rectangle = width × length. Given width = 5, so the factored binomial is length.

Step3: Substitute $n=2$ into $10n+35$

Calculate value:
$10(2)+35 = 20+35 = 55$

Step4: Substitute $n=2$ into factored expression

Calculate value:
$5(2(2)+7) = 5(4+7) = 5×11 = 55$

Answer:

(a) $5(2n+7)$
(b) The area of a rectangle is equal to the product of its width and length. Since the width is 5 feet, factoring the area expression $10n+35$ gives $5(2n+7)$, so $2n+7$ must represent the length.
(c) For $10n+35$ when $n=2$: $10(2)+35=55$. For the expression from (a) when $n=2$: $5(2(2)+7)=55$. Both equal 55, so they are equivalent.