Kaylee is painting a design on the floor of a recreation room using equilateral triangles. She begins by painting the outline of Triangle 1 measuring 50 inches on a side. Next, she paints the outline of Triangle 2 inside the first triangle. The side length of Triangle 2 is 80% of the length of Triangle 1. She continues painting triangles inside triangles using the 80% reduction factor. Which triangle will first have a side length of less than 29 inches?
We can use a geometric sequence to find the side lengths of each successive triangle. The first term is 50 inches and the common ratio is 0.8.
The nth term of the sequence is given by:
a_n = a_1 * r^(n-1)
where a_1 = 50 and r = 0.8.
We want to find the smallest n such that a_n < 29.
29 = 50 * 0.8^(n-1)
dividing by 50:
0.58 = 0.8^(n-1)
taking ln of both sides:
ln(0.58) = (n-1)ln(0.8)
n-1 = ln(0.58) / ln(0.8)
n = 10.49
Since n must be a whole number, the first triangle to have a side length less than 29 inches is the 11th triangle.