A famous fractal called the Sierpinski Triangle starts with one black triangle. The triangle is then cut into four equal pieces, and the three outer triangles are coloured black and the inner triangle is white. Each of the three black triangles is again divided into four equal triangles, and the three outer triangles are shaded black. a. Suppose the area of the first triangle is 100 cm2. Determine the black shaded area in the second triangle. Do the same for the third triangle. b. Write the general term describing the shaded area of the nth Sierpinski Triangle. c. Suppose the diagram continued on. What total area would be shaded in the first ten Sierpinski Triangles? Round to the nearest whole number. d. Suppose the fractal continued forever. Determine the total area shaded in all the
6 years ago
Answered By Chris Y
This is a question about geometric series. Changes by multiples.
We start with 100cm2 area with the whole triangle. The four equal pieces they describe dividing the our initial triangle into resemble the pattern of the ‘tri-force’ if you’re having difficulty imagining what is going on. So after the first step where we divide our 1 all black triangle into 4 equal smaller parts and make the middle white and leave the edges as black. We now only have 75 cm of black. Pictures are very helpful for this next one to be sure. If you divide all 4 triangles again into four equally smaller triangles (we aren’t changing the middle white triangle just dividing it so that we have equal shapes) When we do this we will have our first triangle divided into 16 equal parts. The white middle part from the first step counts as 4 of the 16, then each of the corner black triangles will contribute 3 more of the 1/16’ths. This gives us a total shaded area for this stage of 7 white (middle=4 +3 from the corners) and the remaining part of the 16 is 9 black. So if we look at our pattern it should look like this n is the itteration we are doing, and A is the area
n=1 A=100cm2
n=2 A=75cm2
n=3 A=56.25cm2
Algebraically we would have thisn=1 A=100*(1)
n=2 A=100*(3/4)
n=3 A=100*(9/16)
From this we can generalize a formula
An=100cm2 (3/4)n-1
The question asks us what the total area would be shaded in the first 10 S. Triangles so we plug the number 10 (this question states that the first s. Triangle is the completely black starting point) into our generalized formula and we getA10=100(3/4)9A10=7.5085cm2