A coffeepot has the shape of a cylinder with radius 5 inches, as shown in the figure above. Let h be the depth of the coffee in the pot, measured in inches, where h is a function of time t, measured in seconds. The volume V of coffee in the pot is changing at the rate of -5P~ cubic inches per second. (The volume V of a cylinder with radius r and height h is V - 7rr2h. ) (a) Show that ~ = -5~ . (b) Given that h - 17 at time t = O, solve the differential equation ~ = -5~ for h as a function of t. (c) At what time t is the coffeepot empty?
[ I : dV , -57r~ dt 3 : I : computes ~ 1 : shows result
(a) V - 257rh
1 : separates variables 1 : antiderivatives 1 : constant of integration 1 : uses initial condition h - 17 when t - O 1 : solves for h