Day 5 after-lunch quiz:

1.  Quickly write down answers to each of the following:

• How do you find the volume of a cylinder?

• How do you find the surface area of a cube?

• How do you find the area of a trapezoid?

• How do you find the volume of a pyramid?

• What is the definition of rate of change?

• How do you visualize how many cubic centimeters there are in a liter?

2.  The weight of a gram of water is about 1000 dynes. 

• If you multiply the weight of water exiting the tube by the change in  altitude from water surface to exit point, you get the change in its gravitational potential energy. 
• If you multiply its mass in grams by its exit velocity in centimeters / second, you get its kinetic energy in ergs. 

For your data:

• What is the change in the potential energy of a gram of water? 
• What is the kinetic energy of a gram of exiting water? 

An erg is a dyne * centimeter. 

• How much of the potential energy loss went into kinetic energy? 

The difference is the energy dissipated against friction as the water runs through the tube.

• How much energy was dissipated against friction in the tube?

3.  If the before-transition numbers of sane and demented people are denoted respectively by S and D, and if in a transition 10% of the sane become demented and 20% of the demented become sane, then how many sane and how many demented will there be after the transition?  How many will there be after one more transition?

4.  On a 6 x 7 grid of square tiles:

• How many corners are there to measure between?  That is, if you put a dot at each corner, how many dots would there be?

• Draw the grid and sketch all possible corner-to-corner distances, sketching each possible distance exactly once.

• Is there any distance in your sketch that couldn't be represented by a corner-to-corner line starting from the lower left-hand corner of the grid?

• Place the lowest possible upper limit you can on how many distances are possible between corners of a 10 x 10 grid.  That is, find the smallest number you can that you're sure is at least as great as the number of total distances.  It isn't sufficient to just write down the number that, in your opinion, is the answer.  You need to explain how you reasoned out your answer.

• Place the largest lower limit you can on the number of possible distances.  Naturally this also needs to be explained and justified.

• List all the corner-to-corner distances possible for this grid.

5.  How many intact spheres of diameter 1 centimeter could be fit into a 1-meter cube?

You're unlikely to be able to give the actual number, so place the lowest upper limit you can on the number, and the greatest lower limit you can be sure of.  Explain your reasoning.