§3.1 Related rates (continued)

Exercises (continued)

Summary

With $\frac{dx}{dt} = 3$ and $\frac{ds}{dt} = 2$, the tip of the shadow sits at $y = x + s$. Its speed $\frac{dy}{dt}$ equals

A rope runs from a boat's bow over a pulley $5$ feet higher, with rope length $z$ and horizontal distance $x$, so $x^2 + 25 = z^2$. The rope is pulled in at $2$ feet per second. When $x = 12$ we have $z = 13$, and the boat approaches the dock at

As the skateboarder moves away from the lamppost, the rate $\frac{ds}{dt}$ of his shadow's lengthening

With $3s = 2x$ and $\frac{dx}{dt} = 3$ feet per second, compute $\frac{ds}{dt}$ in feet per second.

Sand falls at $10$ cubic feet per minute onto a conical pile with radius always equal to height. When the pile volume reaches $1000$ cubic feet, $\frac{dh}{dt} = \frac{10}{\pi h^2}$ with $h = \sqrt[3]{\frac{3000}{\pi}}$. Compute $\frac{dh}{dt}$, rounded to four decimals.

In the sailboat problem, the rope length $z$ satisfies $\frac{dz}{dt} = -2$ feet per second because the rope is being pulled in.