Sliding Ladder $(11$ points$)$

A $13$ ft ladder rests against the side of the house. When the base of the ladder is $12$ ft away from the base of the house, it is moving with a velocity of $3\frac{ft}{sec}$ away from the house.

Determine:

a$)$ how fast the top of the ladder is moving $(7$ points$)$

b$)$ the rate at which the area under the ladder is changing $(4$ points$).$

The Melting Snowball $(7$ points$)$

Suppose a spherical snowball is melting and the radius is decreasing at a constant rate, changing from $15$ inches to $9$ inches in $30$ minutes. How fast was the volume changing when the radius was $11$ inches?

Angle of Elevation $(7$ points$)$

A camera a ground level is filming the lift$-$off of a space shuttle that is rising vertically according to the position equation $s=50t^2,$ where $s$ is measured in feet and $t$ is measured in seconds. The camera is $2500$ feet from the launch pad. Find the rate of change in the angle of elevation of the camera at $10$ seconds after lift$-$off.

Hole Lotta Can $(7$ points$)$

An aluminum cylindrical can full of a liquid has sprung a leak of $10\frac{i{n}^3}{(min)}.$ The can has a height of $10$ in $.$ and a diameter of $4$ in $.$ How fast is the height of the liquid falling?

Flying a Kite $(7$ points$)$

A kite is moving horizontally with a speed of $7\frac{(m)}{sec}$ at an altitude of $30$ m $.$ Determine how fast the string is being let out when the kite is $50$ m from the kite flyer.

Leaking Sand $(7$ points$)$

As sand leaks out of hole in a container, it forms a conical pile whose altitude is always the same as the radius. If the height of the pile is increasing at a rate of $6i\frac{n}{m}in,$ find the rate at which the sand is leaking out when the altitude is $12$ inches.