12 If we study the physics of a deflating balloon, taking into consideration the pressure (18 marks) inside the balloon, and the flow of air through the opening (according to the Hagen-Poiseuille equation), we get the differential equation dV dt V 3 where V(t) is the volume of the balloon in cm', t seconds after it started to deflate, and the balloon is assumed to be completely deflated when V = 1 cm3. Let us assume that initially we inflate the balloon to 27 cm3, so V (0) = 27. (a) (5 marks) Find a linear approximation of the solution near t = 0. Hint: What is V'(0)?(c) (3 marks) Below is a slope field for this differential equation. 120 Sketch V (t), not its linear approximation, directly on the slope field. Make sure it's clearly visible. (d) (2 marks) Using the slope field above and your sketch, estimate the time it will take for the balloon to be completely deflated. Time ~ seconds