Calculus/Related Rates/Solutions

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

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:

${\dot {r}}={\frac {100}{4\pi 4^{2}}}=\mathbf {{\frac {25}{16\pi }}{\frac {ft}{min}}}$

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

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:

${\dot {h}}={\frac {(4)(3)}{\pi 6^{2}}}=\mathbf {{\frac {1}{3\pi }}{\frac {ft}{min}}}$

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?