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Writing Prompts Surfaces of Equal Gold Coat Previously, we saw two solids of revolution (a circular cylinder and a sphere) such that when sliced into pieces of equal width, eac h piece has equal surface area. In other words, the surface area of each piece depends only on the width of the piece. This problem seeks to find descriptions of other possible surfaces of revolution that have this property. Let y = f(t) be a curve above the t - axis defined for all t - values. Rotate this curve around the t - axis to create a surface. Consider the part of t he solid in the interval [a, a+h]. We want the solid to meet the condition that the amount of surface area contained in the slice [a, a+h] is proportional to only h and doesn’t depend on a, which slice we take. The function shown in the image below does not satisfy this property because the blue lice on the left has less surface area than that green slice on the right. In symbols, we seek to solve the integral equation where k>0 is the proportionality constant. 1. Differentiate both sides of this equation with respect to h. You’ll use the Fundamental Theorem on the right side. 2. You should see te rms like f(a+h) and f’(a+h) in your result from above. To make the computations simpler, substitute x=a+h. You can think of this as shifting the section of your surface to the origin. Now you have a separable differential equation. One function we kn ow has this property is the constant function f(x) = b. Find the value of k so that f(x) = b is a solution. How does this value of k relate to your answer from the previous WH question? Based on your answer from the previous WH, what would you guess is the value of k for which is a solution? 3. We will search for other possible surfaces. In your separable differential equation, it may be helpful to write f(x) = y and f’(x) = y’ for simplicity. Separate and solve this differential equation for y. Your result will depend on k and C, an integrat ion constant. 4. Plot these solutions with sliders for k and C into Desmos and show a few solutions for various values of k and C. What hypothesis do you make about the family of curves that have this “equal gold coat” property? 5. Show that the solution to your different ial equation can be algebraically rearranged to prove your hypothesis from part 4.