1. Suppose that the volume V of a certain cone can be expressed as V =nh, where h is the height of the cone. a) By differentiating V = nh with respect to time t, derive an equation that relates h, dh/dt, and dV/dt as the cone shrinks. b) At a certain moment, the height of the cone is decreasing at 2 centimeters per second (cm/sec), and its height is 6 cm. At that same moment, what is the rate of change of the volume of the cone? Do not round your answer. Use the correct unit for your answer. c) At a certain moment, the height of the cone is 8 centimeters, and its volume is decreasing at 160 cm /sec. At that same moment, what is the rate of change of the height of the cone? Do not round your answer. Use the correct unit for your answer.

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…

For many species of fish, the allometric relationship between the weight W and the length L is approximately  W = kL3,  where k is a constant. Find the rate of change of the weight as a corresponding rate of change of the length. (dL/dt)

Consider a cylindrical fuel tank of radius r and length L (in some appropriate units) that is lying on its side. Suppose that fuel is being pumped into the tank at a rate q. At what rate is the fuel level rising? Compute the rate of change of the height of fuel when r is 7 m, L is 20 m and q is 12m³/s. ☐ L