Hello, Juggalomike!

1. A light is affixed to the top of a 12 foot tall lamppost.

A 6-foot tall man walks away from the lamp post at a rate of 5 ft/sec.

How fast is the length of his shadow increasing when he is 5 feet away? Code:

A
*
| * C
| *
12 | | *
| 6 | *
| | *
* - - - * - - - - - - - * - - -
B x D s E

The lampost is:

The man is:

His distance from the lampost is: , and

The length of his shadow is:

Since

Differentiate with respect to time: .

Therefore: .

His shadow is lengthening at 5 ft/min.

2. A water tank has the shape of an inverted right circular cone

of altitude 12 feet and base radius 6 feet.

If water is being pumped into the tank at a rate of 10 gal/min,

at what rate is the water level rising when the water is 3 feet deep? Nasty! .Nasty! .They give the volume in __gallons__, then ask for answers in *feet*.

. . They could have given us a conversion formula at the very least!

I had to look it up: .

So we have: .

Look at the side view of the cone: Code:

A 6 D
- * - - - - * - - - - *
: \ | /
: \ | /
: \ B| r /
: * - - + - - * E
12 \ | /
: \ | /
: \ h| /
: \ | /
: \|/
- *
C

The height of the cone is:

The radius of the cone is:

The height of the water is:

The radius of the water is:

Since

The volume of the water is: .

Substitute [1]: .

Differentiate with respect to time: .

Substitute our known values: .

Therefore: .