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 9:S = S:T = 3 and
S and T are the side lengths of the shaded
squares, what is the total shaded area?

Answer 1

Let's denote the side length of the large shaded square as L and the side length of the small shaded square as S.

Since the ratio of L:S is 3, we have L = 3S.
Similarly, since the ratio of S:T is 3, we have S = 3T.

We know that the side length of the large shaded square is 9 feet, so L = 9. Substituting this into the equation L = 3S, we get 9 = 3S, so S = 3 feet.

Substituting this value of S into the equation S = 3T, we get 3 = 3T, so T = 1 foot.

The total shaded area is the area of the large shaded square plus the sum of the areas of the 8 small shaded squares.

Area of large shaded square = L^2 = 9^2 = 81 square feet
Area of small shaded square = S^2 = 3^2 = 9 square feet

Total shaded area = 81 + 8(9) = 81 + 72 = 153 square feet

Therefore, the total shaded area is 153 square feet.