4.4+Cooling+Water

January 13, 2009: AD
Big Idea: The rate at which many things grow or decay can often be described mathematically. Essential Question: How can I model a pattern of repeated division?

__Problem 4.4: Cooling Water__
Given below is the table with our observations. The first one uses Celsius, while the second one is in Fahrenheit.
 * Time (Minutes) || Water temperature (Celsius) || Room temperature (Celsius) || Difference (Room and Water Temp.) ||
 * 0 || 86 || 25 || 61 degrees ||
 * 5 || 76.5 || 25 || 51.5 ||
 * 10 || 71.5 || 26 || 45.5 ||
 * 15 || 66 || 25.5 || 40.5 ||
 * 20 || 61.5 || 26 || 35.5 ||
 * 25 || 58.5 || 25.5 || 33 ||
 * 30 || 55 || 26 || 29 ||
 * 35 || 52 || 25 || 27 ||
 * 40 || 50 || 25 || 25 ||
 * 45 || 48 || 25 || 23 ||
 * Time (Minutes) || Water temperature (Fahrenheit) || Room temperature (Fahrenheit) || Difference (Room and Water Temp.) ||
 * 0 || 184 || 79 || 105 ||
 * 5 || 168 || 78 || 90 ||
 * 10 || 156 || 78 || 78 ||
 * 15 || 148 || 78 || 70 ||
 * 20 || 140 || 78 || 62 ||
 * 25 || 134 || 78 || 56 ||
 * 30 || 128 || 78.5 || 49.5 ||
 * 35 || 124 || 78 || 46 ||
 * 40 || 118 || 77 || 41 ||
 * 45 || 117 || 77 || 40 ||


 * A. Make a graph of your (time, water temperature) data.**

The water temperature changed rapidly in the first few intervals, but then beacme progressively slower. In the graph, we can se that the steepness of the lines connecting the data points becaome shallower and shallower over time, which shows that there was less change in temperature during those intervals. The lines are steepest in the first few intervals, and are shallowest in the last intervals, which proves that the temperature changed rapidly in the beginning and then slowed down.
 * B. Describe the pattern of change in the (time, water temperature) data. When did the water temperature change most rapidly? When did it change most slowly?**

(Look at tables shown above)
 * C. Add a column to your table. In this column, record the difference between the water temperature and the room temperature for each time value.**


 * D. Make a graph of the (time, temperature difference) data.**

As you can see, the shapes of the fahrenheit and celsius graphs correspond almost exactly. The curves look exactly the same, and the steepness of the lines between the data points look very similar.
 * E. Compare the shapes of the graphs.**

As was the case with the (time, water temperature) graph, the temperature difference changes most in the beginning, and then slows down. This is once again shown by the decreasing steepness of the lines connecting the data points.
 * F. Describe the pattern of change in the (temperature, time difference) data.When did the temperature change most rapidly? When did it change most slowly?**

Calculations: 61 / 51.5 = 1.184 51.5 / 45.5 = 1.132 45.5 / 40.5 = 1.123 40.5 / 35.5 = 1.141 35.5 / 33 = 1.076 33 / 29 = 1.138 29 / 27 = 1.074 27 / 25 = 1.08 25 / 23 = 1.087 Average growth factor (as time decreases) = 1.115 Therefore the average decay factor would be the reciprocal of 1.115 which is 1000/1115 = 0.89686 Therefore the decay factor is 0.089686
 * G. Assume that the relationship between the temperature difference and the time in this experiemtn is exponential. Estimate the decay factor for this experiment. Explain how you made you estimate.**


 * H. Find an equation of the (time, temperature difference) data. Your equation should allow you to predict the temperature difference at the end of any 5-minute interval.**

The general form for an exponential equation is y=a(b^x) Starting value (a): 61 Decay Factor (b): 0.89686 x is the number of 5-minute intervals, and y is the temperature difference after x intervals Thus we end up with this equation: y = 61(0.89686^x)

__Problem 4.4 Follow-Up__

If the (time, temperature difference) had continued, the general curve would probably have continued. The lines connecting the data points would have become shallower and shallower until the slope became very close to 0, at which point it would seem as if the curve has straightened.
 * 1. What do you think the graph o the (time, temperature difference) data would look like if you continued the experiment for several more hours?**

Factors like how close the cup of liquid is from hot or cold devices (the air-conditioner for example) can affect how fast the liquid cools.
 * 2. What factors might affect the rate at which a cup of hot liquid cools?**

Factors such as mulfunctioning thermometers, improper readings, or external factors (such as ACs) can introduce errors in the data as it could lead to faster cooling, figures that dont make sense with the situation, etc.
 * 3. What factors might introduce errors in the data you collect?**