265 Project 1: Koch-ing up Mackerel Name: 1. (15 points) The Koch Snowflake has an infinite perimeter but finite area. ˆ The first step is to draw an equilateral triangle. Then on each side, draw another equilateral triangle one-third of the size. Continue. Here is a picture after two steps with the first piece of the third step included. Finish the third step and add the fourth step. ˆ Determine the area of the shape you have drawn if the length of the original triangle is 3 inches. Include units. ˆ The Koch Snowflake results after an infinite number of iterations. Construct a geometric series that represents the area of the Koch Snowflake. ˆ Compute the sum of the geometric series you constructed. 2. (15 points ) Suppose the P (t) models a population of mackerel measured in thousands of kilograms of biomass at time t in years. P (t) satisfies dP = 2P − P 2 where P (0) = 3. dt ˆ Determine P 00 (t) by taking the derivative of the differential equation using implicit differentiation. That means dtd P 2 = 2P P 0 . ˆ Get P 000 (t) in the same way. ˆ Use the above derivatives and the equation, itself, to construct a third-degree Taylor polynomial for P (t). ˆ Approximate P (0.1) using your Taylor polynomial. 2t ˆ The explicit form of P (t) is 3e6e2t −1 . Ascertain the error between the Taylor polynomial and the explicit form at t = 0.1. 2