k=0 AAAAA k=1 k=2 k=3 The process pictured above starts with an equilateral triangle and proceeds by recursively removing equilateral triangles as shown in the figure. a) If we begin with an equilateral triangle of area A(0) = 3, what is the area of the shaded region after & steps of removing triangles? A(k) = b) What is the shaded area in the limit as k o? A = lim A(k) = k-00 c) What is the total shaded area shown in all the stages? A(k) = k=0 d) If we begin with an equilateral triangle of side length 2, let P(k) be the perimeter of the shaded region after k steps. Calculate P(0)= P(1) = P(2) = P(3) = Now calculate the perimeter of the shaded region after steps. P(k) = e) What is the perimeter in the limit as k o? P = lim P(k) = k-xx Note: Permitted inputs for limits or series include numbers, "inf", "-inf", and "DNE". k=4 k=5 |