Suppose a raindrop evaporates as it falls but maintains its spherical shape. Assume that the rate at which the raindrop evaporates (that is, the rate at which it loses mass) is proportional to its surface area, where the constant of proportionality is –0.01. The density (mass per volume) of water at 3.98°C is 1 g/cm3. The surface area of a sphere is 4πr2, and its volume is 4πr3/3, where r is the radius. Assume no air resistance. (Project 8 models the motion of this raindrop under the influence of air resistance.) Assume that the initial radius is 0.3 cm. Determine the raindrop’s initial mass. Write a differential equation for the rate of change of mass as a function of r. Write an equation for r as a function of mass

At the bottom of a swimming pool, the pressure exerted at a given point is proportional to the depth of the water above it. In particular, the pressure at a given point of depth d is equal to 10, 000d Pascals. We will investigate the total pressure at the bottom of a swimming pool depending on its shape. You may view 3D models of the swimming pools for reference here (note that this is one model plotting the depth as a function of x and y, so represents essentially the pool flipped upside-down): https://www.geogebra.org/m/nnf4yhax (a) Suppose we have a rectangular pool that is 50 meters long and 25 meters wide. The pool has a shallow end that is 1 meter deep at the edge, a deep end 50 meters away that is 3 meters deep at the edge, and is such that the bottom of the pool forms a linear slant in between (so along a line segment parallel to the short side, the pool has constant depth, and along a line segment parallel to the long side, the pool's depth varies linearly). Set up and…

Suppose that a motorboat is moving at 42 ft/s when its motor suddenly quits, and that 11 s later the boat has slowed to 23 ft/s. Assume that the resistance it encounters while coasting is proportional to its velocity, so that dv/dt = - kv. How far will the boat coast in all?