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Cooling Water And Exponential Funtions Essay

979 words - 4 pages

The Water Cooling ExperimentDuring this lab we measured, using our CBL's and graphing calculators, the change in temperature of a glass of boiling water left to cool in a room. The purpose of the lab is to see how exponential functions can be applied to real-life applications. During the experiment I'm expecting for the water temperature to decrease as soon as the water is taken of off the heating device. If the water is left in the room long enough, the temperature will be able to reach room temperature, no less.Using the calculator, CBL, and temperature probe, the room temperature was taken to find out the lowest temperature the water could cool to (the asymptote). To get the most accurate room temperature reading, I had to keep the probe extremely still because movement could cool off the probe and give a false reading. Also by shortening the intervals in which the CBL took the readings allowed for accurate results. A glass of water was then brought up to boiling temperature. After the water started boiling, a temperature probe (connected to the CBL) was left in the water as the calculator and CBL took temperature readings. I let the probe sit in the water for a few seconds before starting the temperature readings to allow the probe to heat up to the same temperature of the water. This prevented the results from showing an increase and then decrease in temperature. The readings were taken every 60 seconds for 36 minutes.The data, as stated before, was expected to resemble an exponential decay function. The graph reinforced this hypothesis by showing an original steep decrease in temperature, and then slowing down as it neared the asymptote (room temperature) at 22.11 degrees Celcius (see figure 1). This data set was saved to list 1 and list 2 in the calculator. List 3 and 4 consisted of the log of x and y respectively (see spreadsheet - figure 4). A linear regression line was then performed on (x, log(y)) and (log(x), log(y)). As expected the linear regression of (x, log(y)), the exponential model, was the better fit. We know this because the absolute value of the "corr" value was closer 1 than the (log(x),log(y)) power function's (see figure 5). Once the linear equation was found it needed to be changed into an exponential equation:When this equation is graphed along with the data, I found that the line looked to be as good of a fit as the "corr" coefficient showed it would be (see figure 2). To see exactly how good of a fit the line was, I found the residuals by substituting the original x values into the exponential equation and then subtracting the y corresponding y values: y1(x)-y. These residuals show you how far away from the line the data set is. The graph of the residuals (see figure 3) shows that the line is a good fit because the points...

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