## Fun Related Rates / Optimization Question: Smallest Surface Area of a Square and Circle Cut From a Single Piece of Rope

Hello everyone, One of my Calculus students had an interesting Related Rates problem that I had to go home and think about for a while in order to figure out. The problem was set up as such: A 25 inch piece of rope needs to be cut into 2 pieces to form a square and a circle. How should the rope be cut so that the combined surface area of the circle and square is as small as possible? Here's what we'll need to do: 1. We will have to form equations that relate the length of the perimeter and circumference to the combined surface area. 2. We will then differentiate to create an equation with the derivative of the surface area with respect to lengths of rope. 3. Wherever this derivative equals 0 there will be a maxima or minima, and so we will set the derivative = to 0 and determine which critical points are minima... read more