MHB What Is the Ratio of Area to Perimeter for a Rug with Length 9w?

[Kathleen217744]

The length of a rug is eight times greater then the width. if the width of the rug is (w+5), what is the ratio of the area of the rug to the perimeter of the rug in simplest form?

 

[Joppy]

Re: help

Hi, welcome to the forum!

Let's write out what we know.

$width = w + 5$

The length is also given, but it's been disguised a little bit.

The length of a rug is eight times greater then the width

This just translates to, $length = 8 \cdot width = 8 \cdot (w + 5)$.

Now we have everything we need from the question, let's look at the next part.

what is the ratio of the area of the rug to the perimeter of the rug in simplest form?

Let's focus first on finding the area of the rug, and the perimeter of the rug.

area of rug $= A = length \cdot width = 8(w + 5)(w + 5) = 8(w + 5)^2$.

perimeter of rug $= P = 2 \cdot length + 2 \cdot width = 2w + 10 + 16w + 80 = 18w + 90 = 18(w + 5)$.

Finally, the question asks for the ratio of the area to the perimeter. Let's call the ratio R.

Then, $R = A:P = \dfrac{A}{P} = \dfrac{8(w + 5)^2}{18(w + 5)^1}$

Can you finish it off? :)

 

[HallsofIvy]

"8 times the width" would be 8w but I would interpret "8 times greater than the width" as "the width plus 8 times the width"- w+ 8w= 9w.