Ex 8.3, 7

Transcript

Ex 8.3, 7 Fig. 8.12 shows Stages 0 to 3 of the Sierpiński square carpet. Stage 0 of this fractal is a square sheet of paper. To construct Stage 1, each side of the square is trisected and the points of trisection of opposite sides are joined to obtain nine smaller squares. The centre square is then removed and the 8 smaller squares are retained, leaving a square hole in the centre. The same process is repeated on the eight smaller shaded squares to obtain Stage 2 and so on. Ex 8.3, 7 (i) How many red squares are there in Stages 0 to 3? At each stage, every existing square is split into 9 smaller squares, and the middle one is removed. This means we keep 8 squares for every 1 square we had previously. Stage 0: 1 Stage 1: 8 Stage 2: 8 × 8 = 64 Stage 3: 64 × 8 = 512 Ex 8.3, 7 (ii) Can you predict the number of red squares in Stages 4 and 5? We just continue multiplying by our common ratio of 8. Stage 4: 512 × 8 = 4,096 squares Stage 5: 4096 × 8 = 32,768 squares Ex 8.3, 7 (iii) Can you find a rule for the number of red squares at the 𝑛^"th " stage? Write the explicit formula as well as the recursive formula for the number of red squares at any stage. Explicit Formula Because we are multiplying by 8 at each stage (starting at Stage 0), the stage number is the exponent. The explicit rule is 𝒕_𝒏=𝟖^𝒏 Recursive Formula We start with 1 square, and to get to the next stage, we multiply the previous number by 8 . The recursive rule is 𝒕_𝟎=𝟏,𝒕_𝒏=𝟖×𝒕_(𝒏−𝟏) (for 𝒏≥𝟏) Ex 8.3, 7 (iv) Suppose the area of the square in Stage 0 is 1 square unit. What is the area of the red region in Stages 1, 2 and 3? What will be the area of the red region in Stages 4 and 5? Find the explicit as well as the recursive formula for the area of the red region at the 𝑛^"th " stage. What happens to this area as 𝑛, the number of stages, goes on increasing? Let’s look at our figure again Ex 8.3, 7 (iv) Suppose the area of the square in Stage 0 is 1 square unit. What is the area of the red region in Stages 1, 2 and 3? What will be the area of the red region in Stages 4 and 5? Find the explicit as well as the recursive formula for the area of the red region at the 𝑛^"th " stage. What happens to this area as 𝑛, the number of stages, goes on increasing? Let’s look at our figure again If the original area is 1 square unit, at Stage 1, we divide it into 9 pieces and throw 1 away. This means we keep 𝟖/𝟗 of the area. This is a shrinking GP where 𝒓=𝟖/𝟗. Now, finding Area of different stages Stage 1 Area: 1 × 8/9=8/9 Stage 2 Area: 8/9 ×8/9=(8/9)^2=64/81 Stage 3 Area: (8/9)^2 ×8/9=(8/9)^3=512/729 Stage 4 Area: (8/9)^3 ×8/9=(8/9)^4 Stage 5 Area: (8/9)^4 ×8/9=(8/9)^5 Thus, Explicit Formula: 𝑨_𝒏=(𝟖/𝟗)^𝒏 Recursive Formula: 𝑨_𝟎=𝟏,𝑨_𝒏=𝟖/𝟗×𝑨_(𝒏−𝟏) (for 𝒏≥𝟏 ). We are also asked What happens to this area as 𝒏 goes on increasing? Just like the Sierpiński triangle, as you zoom in forever, the number of red squares approaches infinity, but the total area of those squares steadily shrinks until it gets closer and closer to 𝟎. Stage 5 Area: (8/9)^4 ×8/9=(8/9)^5 We are also asked Explicit and Recursive formula Explicit Formula: 𝑨_𝒏=(𝟖/𝟗)^𝒏 Recursive Formula: 𝑨_𝟎=𝟏,𝑨_𝒏=𝟖/𝟗×𝑨_(𝒏−𝟏) (for 𝒏≥𝟏 ). Finally, we are asked What happens to this area as 𝒏 goes on increasing? Just like the Sierpiński triangle, as you zoom in forever, the number of red squares approaches infinity, but the total area of those squares steadily shrinks until it gets closer and closer to 𝟎.