What do I have to do to solve the problem?

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 64 inches on a side. Next, she paints the outline of Triangle 2 inside the first triangle. The side length of Triangle 2 is 90% of the length of Triangle 1. She continues painting triangles inside triangles using the 90% reduction factor. Which triangle will first have a side length of less than 38 inches?

Do you know what an equilateral triangle is?

Each triangle she paints is 90% the size of the previous one. So the first triangle has length 64, the second is 64*90%, the third is 64*90%*90%, the fourth is 64*90%*90%*90%, etc.

We can write this mathematically as

l_n=64\cdot 0.9^{n-1}, where l_n is the length of the nth triangle.

Note that we need to use (n-1) in the exponent because the first triangle was full-sized, the second triangle is only multiplied by 90% once, the third triangle is only multiplied twice, etc.

Now that we know the length of the nth triangle, we can just set l_n=38 and solve to find the value of n. If we solve like this, it's likely that n won't turn out to be a perfect integer. That's okay; it just means that whatever the next higher integer is after our non-integer calculated value is the minimum number she'd have to paint before the length of a side got smaller than 38".

So now let's solve the equation:

64\cdot 0.9^{n-1}=38
0.9^{n-1}=\frac{19}{32}
\log _{0.9} 0.9^{n-1}=\log _{0.9} \frac{19}{32}
n-1=\log _{0.9} \frac{19}{32}
n=\log _{0.9} \frac{19}{32}+1
n=\frac{\log \frac{19}{32}}{\log 0.9}+1
n\approx 5.948

So since the answer is somewhere between 5 and 6, we know that the 5th triangle won't be quite small enough, and the 6th will be slightly smaller than 38". So the answer to the word problem is the 6th triangle.

Now let's check just to be sure:

5th triangle: l_5=64\cdot 0.9^{5-1}=41.9904"
6th triangle: l_6=64\cdot 0.9^{6-1}=37.79136"

Sure enough, it's the 6th triangle before the length is less than 38".

Hopefully that makes sense.

Sorry the math didn't get spaced very well. Hopefully you can still read it. I hate that you can't preview your answers on this forum.