# Calculus Homework #8 Due 4-20

1. A spherical snowball is melting. Its radius is decreasing at 0.2 cm per hour when the radius is 15 cm. How fast is its volume decreasing at that time?

2. A  potter forms a piece of clay into a cylinder. As he rolls it, the length, $L$, of the cylinder increases and the radius, $r$, decreases. If the length of the cylinder is increasing at 0.1 cm per second, find the rate at which the radius is changing when the radius is 1 cm and the length is 5 cm.

3. A certain quantity of gas occupies a volume of 20cc at a pressure of 1 atmosphere. The gas expands without the addition of heat, so, for some constant $k$, its pressure, $P$, and volume, $V$, satisfy the equation $PV^{1.4}=k$. (a) Find the rate of change of pressure with respect to volume. (b) The volume is increasing at 2cc/min when the volume is 30cc. At that moment, is the pressure increasing or decreasing? How fast?

4. (a) A hemispherical bowl of radius 10cm contains water to a depth of $h$cm. Find the radius of the surface of the water as a function of $h$. (b) The water level drops at a rate of 0.1 cm per hour. At the what rate is the radius of the water decreasing when the depth is 5 cm?

1. Find the exact global minimum value of $f(x)=x+\frac{1}{x}$ on the domain $\{x|x>0\}$
2. Find the exact global minimum and maximum values of $g(t)=te^{-t}$ on the domain $\{t|t\geq0\}$
3. A certain quantity of gas occupies a volume of 20cc at a pressure of 1 atmosphere. The gas expands without the addition of heat, so, for some constant $k$, its pressure, $P$, and volume, $V$, satisfy the equation $PV^{1.4}=k$. (a) Find the rate of change of pressure with respect to volume. (b) The volume is increasing at 2cc/min when the volume is 30cc. At that moment, is the pressure increasing or decreasing? How fast?
4. (a) A hemispherical bowl of radius 10cm contains water to a depth of $h$cm. Find the radius of the surface of the water as a function of $h$. (b) The water level drops at a rate of 0.1 cm per hour. At the what rate is the radius of the water decreasing when the depth is 5 cm?