§3.1 Related rates (continued)

Related Rates Problems

Sand is being dumped by a conveyor belt onto a pile so that the sand forms a right circular cone, as pictured in Figure 3.5. How are the instantaneous rates of change of the sand's volume, height, and radius related to one another?

In the setting of Example 3.1.1, suppose we also know the following: (a) sand falls from the conveyor in such a way that the height of the pile is always half the radius, and (b) sand falls from the conveyor belt at a constant rate of 10 cubic feet per minute. How fast is the height of the sandpile changing at the moment the radius is 4 feet?

A conical tank has depth $8$ feet and top radius $6$ feet. If water stands at depth $h$ with surface radius $r$, similar triangles give

With $r = \frac{3}{4}h$ and $V = \frac{1}{3}\pi r^2 h$, the water volume becomes a function of depth alone, namely

Water enters the tank at $4$ cubic feet per minute, so $\frac{dh}{dt} = \frac{64}{9\pi h^2}$. The water level rises most rapidly

For the tank with $\frac{dV}{dt} = 4$ cubic feet per minute, compute $\frac{dh}{dt}$ when $h = 3$ feet, rounded to four decimals.

A camera $4000$ feet from a rocket's launch pad tracks the rocket, with $h^2 + 4000^2 = z^2$. If the rocket rises at $600$ feet per second at $h = 3000$, compute $\frac{dz}{dt}$ at that instant.

A skateboarder rides at $\frac{dx}{dt} = 3$ feet per second under a lamppost with $3s = 2x$. His shadow length grows at a constant $2$ feet per second.