Chapter 14 Project

 

Part I

 

In this chapter, we discussed Newton's law of cooling. According to Sir Isaac Newton, objects warmer than their surroundings will eventually cool to a common temperature with the surroundings. The cooling rate depends on how much hotter the object is than its surroundings.

 

You will need a Calculator Base laboratory (CBL) with temperature probe, your TI-83 Plus

calculator, a cooler with ice, and boiling water in a cup.

 

1. Enter the COOLTEMP program from the Prentice-Hall companion website.

 

2. Connect the TI-83 Plus calculator to the CBL that has a temperature probe in Chan 1.

 

3. Place the temperature probe in the ice chest and close the lid. Press “mode” to determine the

    temperature of the surroundings inside the ice chest, and record this temperature as T0.

 

4. Place the temperature probe in a cup of boiling water. Allow the temperature probe to

    warm to the temperature of the water.

 

5. Place the cup of water with the probe in the ice chest. Run the program following the directions

    on the screen. Allow the program to run for 15 minutes.

 

The COOLTEMP program is designed to record the temperature the instant the program begins

and every 60 seconds for 30 minutes. It will store in the number of minutes, x, in L3 and the

temperature T in degrees Celsius for L4. The program will also graph the results on your calculator.

 

1.         Write an equation for the recorded temperature, T. Use the STAT CALC command "ExpReg L3, L5

      " to determine an exponential regression model in the form T =  for the data recorded. 

 

2.         Enter the equation from step 1 in Y1, and graph the equation. Does this equation appear to "fit" the recorded                      

             points?

 

3.         According to Newton's law of cooling, the temperature T of a body in t minutes after it is placed in surroundings having a constant temperature T0, is determined by the function , where C is the difference between the initial temperature and the surrounding temperature and k is the cooling constant.

   

a.         Let T0 equal the temperature inside the ice chest, T equal the temperature of the water

      after cooling 30 minutes, and t equal 30 minutes. Determine the constant of cooling, k.

   

b.         Write an equation for T.

 

c.         Store the equation in Y2 and graph it. Does this equation appear to "fit" the recorded points?

 

4. Compare the equations found in step 1 and step 3. Which equation appears to be a better model? Can you explain why the equations differ?

 

Part II

 

In this chapter, we used an exponential function to approximate the population of the United States. The world population may also be determined using an exponential function.

 

1. Search the Internet or your library to find the population of the world from 1991 to the present.

 

2. Enter this data in your calculator. Let L1 be the number of years after 1990 and L2 be the world population.

 

3. Calculate an exponential equation, using the STAT CALC command "ExpReg L1, L2 ".

 

4. Enter the equation found in step 3 in Y1, and graph it. Does this equation appear to "fit" the recorded   

    points?

 

5. Use the equation in step 3 to approximate the world population in 2010.

 

6. Search the Internet or your library to find the projected population of the world in 2010. Compare your results

    in step 5 to the projections. If the results differ, explain why?

 

Part III

 

Many situations may fit into the exponential growth model. Some of these include the spread of the AIDS virus, growth of bacteria, sales of compact discs, sales of computers, and values of collectibles such as art or sport cards. Collect data on a variable that may be experiencing exponential growth. You may search the Internet or your library.

 

1. Develop a mathematical model to represent the growth.

2. Check your model against the data you collected.

3. Use the model to predict a future value of the variable.

 

Write a summary of your findings.

 

Part IV

 

Many situations may fit into a logarithmic model. As you saw in this chapter, the magnitude R of an earthquake of intensity I is defined by the function , where I0 is the intensity of a very small vibration in the Earth used as a standard. Collect data on two major earthquakes. You will need to know the magnitude R measured on the Richter scale for each. Place the magnitude data into the model and solve for the intensity of each earthquake, measured in terms of I0. Then form the ration of the two intensities and interpret the result. How do the two earthquakes compare in their intensities?