Question 5
A sequence of shapes is made as follows.
1. Shape $S_1$ is a shaded square of side 1 unit.
2. Shape $S_2$ is made by dividing $S_1$ into 9 equal squares and removing four of these, so that only the central and corner squares remain.
3. Shape $S_3$ is made by applying the process in step 2 to each of the squares of $S_2$.
4. Shape $S_4$ is made by applying the process in step 2 to each of the squares of $S_3$.
And so on.
(a) Find the area and perimeter of shape $S_3$, giving your answers as fractions.
(b) Find the least value of $k$ for which shape $S_k$ has area less than $1/30$ and also perimeter greater than 30.
1. Shape $S_1$ is a shaded square of side 1 unit.
2. Shape $S_2$ is made by dividing $S_1$ into 9 equal squares and removing four of these, so that only the central and corner squares remain.
3. Shape $S_3$ is made by applying the process in step 2 to each of the squares of $S_2$.
4. Shape $S_4$ is made by applying the process in step 2 to each of the squares of $S_3$.
And so on.
(a) Find the area and perimeter of shape $S_3$, giving your answers as fractions.
(b) Find the least value of $k$ for which shape $S_k$ has area less than $1/30$ and also perimeter greater than 30.