# Calculus/Related Rates/Solutions

1. A spherical balloon is inflated at a rate of $100 ft^3/min$. Assuming the rate of inflation remains constant, how fast is the radius of the balloon increasing at the instant the radius is $4 ft$?

Known:
$V=\frac{4}{3}\pi r^{3}$
$\dot{V}=100$
$r=4$
Take the time derivative:
$\dot{V}=4\pi r^{2}\dot{r}$
Solve for $\dot{r}$:
$\dot{r}=\frac{\dot{V}}{4\pi r^{2}}$
Plug in known values:
$\dot{r}=\frac{100}{4\pi4^{2}}=\mathbf{\frac{25}{16\pi} \frac{ft}{min}}$

2. Water is pumped from a cone shaped reservoir (the vertex is pointed down) $10 ft$ in diameter and $10 ft$ deep at a constant rate of $3 ft^3/min$. How fast is the water level falling when the depth of the water is $6 ft$?

Known:
$h=2r$
$V=\frac{1}{3}\pi r^{2}h=\frac{1}{3}\pi(\frac{h}{2})^{2}h=\frac{1}{12}\pi h^{3}$
$\dot{V}=3$
$h=6$
Take the time derivative:
$\dot{V}=\frac{1}{4}\pi h^{2}\dot{h}$
Solve for $\dot{h}$:
$\dot{h}=\frac{4\dot{V}}{\pi h^{2}}$
Plug in known values:
$\dot{h}=\frac{(4)(3)}{\pi6^{2}}=\mathbf{\frac{1}{3\pi} \frac{ft}{min}}$

3. A boat is pulled into a dock via a rope with one end attached to the bow of a boat and the other wound around a winch that is $2ft$ in diameter. If the winch turns at a constant rate of $2rpm$, how fast is the boat moving toward the dock?

Let $R$ be the number of revolutions made and $s$ be the distance the boat has moved toward the dock.
Known:
$\frac{R}{s}=\frac{1}{2\pi r}$ (each revolution adds one circumferance of distance to s)
$\dot{R}=2$
$r=1$
Solve for $s$:
$s=2\pi rR$
Take the time derivative:
$\dot{s}=2\pi r\dot{R}$
Plug in known values:
$\dot{s}=2\pi(1)(2)=\mathbf{4\pi\frac{ft}{min}}$

4. At time $t=0$ a pump begins filling a cylindrical reservoir with radius 1 meter at a rate of $e^{-t}$ cubic meters per second. At what time is the liquid height increasing at 0.001 meters per second?

Known:
$V=\pi r^{2}h$
$\dot{V}=e^{-t}$
$r=1$
$\dot{h}=0.001$
Take the time derivative:
$\dot{V}=\pi r^{2}\dot{h}$
Plug in the known values:
$e^{-t}=0.001\pi$
Solve for t:
$\mathbf{t=-\ln(.001\pi)}$