Mathematical Modeling 2001

Problems 0627

1. The rate at which water level y changes in a uniform cylinder is -.3 cm / sec * `sqrt( depth in cm), abbreviated by

dy / dt = -.3 `sqrt ( y ).

If the water level is originally y = 100 cm, then

• At what rate will the water level be changing?
• If it continues changing at this rate for 5 seconds what will be the new level?

Starting from the new level:

• At what rate will the water level be changing?

• If it continues changing at this rate for 5 seconds what will be the new level?

Starting from the new level:

• At what rate will the water level be changing?

• If it continues changing at this rate for 5 seconds what will be the new level?

Continue this process for 3 more steps.

Make a table and a graph of water level vs. clock time. What function models your results?

2. If the rate at which the temperature of your thermometer changes is given by rate = -.8 deg / sec * (difference between thermometer temperature and room temperature), abbreviated

dT / dt = -.048 ( T – Tr)

then if the initial temperature is 100 degrees and the room temperature is 50 degrees:

Starting from the initial temperature:

• At what rate will the temperature be changing?
• If it continues changing at this rate for 5 seconds what will be the new temperature?

Starting from the new temperature:

• At what rate will the temperature be changing?
• If it continues changing at this rate for 5 seconds what will be the new temperature?

Repeat this process for three more steps.

Make a table and a graph of temperature excess (this is the excess temperature above room temperature—i.e., subtract room temperature from actual temperature) vs. clock time and see what function models your results.

3. When a pendulum is at displacement x from its equilibrium position, the rate at which its velocity changes with respect to clock time is given by

rate of velocity change = -20 cm/s/s * displacement from equilibrium

which is abbreviated

dv / dt = -20 * x.

If the pendulum is originally 10 cm from its equilibrium position and at rest then:

• At what rate is the velocity changing with respect to clock time?
• If the velocity continues changing at this rate then what will be the velocity after .1 second?
• What then will be the average velocity for this .1 second time interval?
• How much will the position of the pendulum therefore change during the .1 second interval?
• What will be the new displacement of the pendulum?

Starting from the new position:

• At what rate is the velocity changing with respect to clock time?
• If the velocity continues changing at this rate then what will be the velocity after .1 second?
• What then will be the average velocity for this .1 second time interval?
• How much will the position of the pendulum therefore change during the .1 second interval?

• What will be the new displacement of the pendulum?

Repeat this process for three more steps.

Make a table and a graph of pendululm position vs. clock time.

4. A uniform cylinder has radius 4 cm and is initially filled to a depth of 90 cm above a hole .4 cm in diameter. Water flows through the hole at a speed given by v = `sqrt( 2 * 980 * y), where y is the depth in cm and v is the velocity in cm / sec.

Initially:

• At what speed will the water be flowing from the hole?
• How many cubic cm of water will flow from the hole per second?
• By how much must the depth change in a second to accomodate this flow?
• If depth keeps changing at this rate, what will be the depth after 5 seconds?

Starting from this depth:

• At what speed will the water be flowing from the hole?
• How many cubic cm of water will flow from the hole per second?
• By how much must the depth change in a second to accomodate this flow?
• If depth keeps changing at this rate, what will be the depth after 5 seconds?

Repeat this calculation for three more 5-second time intervals. Make a table and sketch a graph of depth vs. clock time.

5. Sketch by hand a graph of rubber band stretch vs. number of milliliters of water. Estimate the equation of the best-fit line by sketching the line that best fits the data and measuring its slope (use points on the line to measure the slope—don't use data points). Is there a pattern to the way your data differ from the straight-line prediction?

Note that stretch is the length in excess of the maximum unstretched length of the rubber band.

6. Sketch by hand a graph of range of water stream vs. of milliliters of water. Estimate the equation of the best-fit line by sketching the line that best fits the data and measuring its slope (use points on the line to measure the slope—don't use data points). Is there a pattern to the way your data differ from the straight-line prediction?

Assuming that the depth vs. clock time function is quadratic, why should we expect the flow range to be linear?

7. What depth function would go with a rate-of-depth-change function y' = dy/dt = .01 t – 3?

What rate function y' would go with a depth function y = .07 t^2 – 3 t + 12?

8. If you had two rubber bands in series—hook, rubber band, hook, rubber band—and redid the experiment, how would your graph of stretch vs. weight be changed?