3. the length of a rectangle is $\frac{w + 3}{w^{2}+w - 12}$ and the width is $\frac{w^{2}+7w +…

3. the length of a rectangle is $\frac{w + 3}{w^{2}+w - 12}$ and the width is $\frac{w^{2}+7w + 12}{w^{2}-9}$. what is the area of the rectangle in simplest form?

3. the length of a rectangle is $\frac{w + 3}{w^{2}+w - 12}$ and the width is $\frac{w^{2}+7w + 12}{w^{2}-9}$. what is the area of the rectangle in simplest form?

Answer

Explanation:

Step1: Recall area formula

The area $A$ of a rectangle is $A = l\times w$, where $l$ is the length and $w$ is the width. Given $l=\frac{w + 3}{w^{2}+w - 12}$ and $w=\frac{w^{2}+7w + 12}{w^{2}-9}$.

Step2: Factor the polynomials

Factor the denominators and numerators:

  • $w^{2}+w - 12=(w + 4)(w-3)$
  • $w^{2}+7w + 12=(w + 3)(w + 4)$
  • $w^{2}-9=(w + 3)(w - 3)$ So, $l=\frac{w + 3}{(w + 4)(w-3)}$ and $w=\frac{(w + 3)(w + 4)}{(w + 3)(w - 3)}$

Step3: Calculate the area

$A=l\times w=\frac{w + 3}{(w + 4)(w-3)}\times\frac{(w + 3)(w + 4)}{(w + 3)(w - 3)}$ Cancel out the common factors: Cancel out $(w + 3)$ and $(w + 4)$ terms. $A=\frac{w + 3}{(w - 3)(w - 3)}=\frac{w+3}{(w - 3)^{2}}$

Answer:

$\frac{w + 3}{(w - 3)^{2}}$