Question

A square carpet of side length 9 feet is designed with one large shaded square and eight smaller, congruent shaded squares, as shown.If the ratios $9:\text{S}$ and $\text{S}:\text{T}$ are both equal to 3 and $\text{S}$ and $\text{T}$ are the side lengths of the shaded squares, what is the total shaded area?

Answer 1

Answer:

$3\sqrt{3} + 72$

Step-by-step explanation:

I attached an image to aid the understanding of the question.

Looking at the image, we see that the 8 shaded parts are congruent, as affirmed in the question as well. And we are told that T = 3, this implies that the area of the square with T as it's side is 9ft². Since all the 8 squares are congruenrt, it means each square has its area to be 9. Therefore, the total area of the 8 shaded squares will be:

$9 \times 8 = 72ft^{2}$

It remains the area of the shaded square with side S.

From the question, we have the following ratio:

$\frac{9}{S} = \frac{S}{T}[\tex]

But

[tex]\frac{9}{S} = \frac{S}{3}[\tex]

Multiplying through by 3S we have

27 = S² and this gives:

[tex]S = \sqrt{27} = 3\sqrt{3}$

I did not add ± because length is always positive, so the case of negative is eliminated.

Now the areas of S is $3\sqrt{3}$

Therefore, the total area of the shaded squares is

$3\sqrt{3} + 72$

Answer 2

Answer:

17

Step-by-step explanation:

We are given that $\frac{9}{\text{S}}=\frac{\text{S}}{\text{T}}=3. \[\frac{9}{\text{S}}=3\]$gives us S=3, so$\[\frac{\text{S}}{\text{T}}=3\]$gives us T=1. There are 8 shaded squares with side length $\text{T}$ and there is 1 shaded square with side length $\text{S},$ so the total shaded area is $8\cdot(1\cdot1)+1\cdot(3\cdot3)=8+9=\boxed{17}.$