Contents. 9. Fractional and Quadratic Equations 2 Example Example Example

Transcription

1 Contents 9. Fractional and Quadratic Equations 2 Example Example Example

2 Peterson, Technical Mathematics, 3rd edition 2 Example 9.52 The average price of residential electricity is shown in Table 9.1. Table 9.1: U.S. Residential Electricity Prices, Year Price ($/kwh) (a) Plot the points. (b) Determine a quadratic regression curve that will fit the data. (c) Use the regression curve to estimate the price of residential electricity in 2. Solutions FIGURE 9.14a (a) Begin by plotting the points on a scatterplot using the procedures outlined in Section 4.6. When using large numbers like years, it is often best to change them to smaller numbers. For example, change 1992 to 92, and 1993 to 93. That way, the number of decimal places in the model will not be as critical. The result is shown in Figure 9.14a. As you can see, the points do not appear to lie on a line. In fact, except for the data for 1997, they look as if they could lie on a curve similar to the one in Figure 9.12b. (b) Since the points seem to lie on a quadratic, we will use quadratic regression. Right click on one of the data points and select Add Trendline. (See Figure 9.14b.) Then select Polynomial and Order 2, as shown in Figure 9.14c, which means the model will be a quadratic. Under Options, choose Display Equation on Chart. The result is shown in Figure 9.14d y = x x FIGURE 9.14b FIGURE 9.14c FIGURE 9.14d FIGURE 9.14e (c) We now need to use this quadratic regression equation to determine the price of residential electricity in the year 2. To predict the price of residential electricity in the year 2, we copy the equation for the model into cell I2, see Figure 9.14e. Enter 1 (for 2) in cell I1 and the result is approximately $8.1/kWh in 2.

3 Peterson, Technical Mathematics, 3rd edition 3 Example 9.54 The data in Table 9.3 shows the consumption of beef in the United States from 194 through (a) Plot the points. (b) Determine a quadratic regression curve that will fit the data. Table 9.3: Beef Consumption in the United States Year (t) Beef consumption ,49 19,611 23,39 25,686 (million pounds) Year (t) Beef consumption 23,56 24,524 24,3 25,534 26,34 26,931 (million pounds) Solutions (a) The points are plotted in Figure 9.17a. (b) These points do not look as if they lie along a straight line. At this stage our only other choice is a parabola. If a quadratic regression is conducted on these points the result is graphed in Figure 9.17b. The regression equation is B(t) 6.699t t million pounds of beef consumed in year t after FIGURE 9.17a y = x x FIGURE 9.17b

4 Peterson, Technical Mathematics, 3rd edition 4 Example 9.55 Use a piecewise-defined function to model the data in Table 9.3. Solution In order to perform a different regression on two parts of the same table, we must enter the data for the graph as two separate series. The obvious place to separate the graph into two functions is after year 197. This is indicated by the vertical line in Figure 9.18a FIGURE 9.18a Enter the data as shown in Figure 9.18b. The first step is to highlight only the data for the first series, the years 194 through 197. Using only those points, construct a scatter plot as shown in Figure c. FIGURE 9.18b Then right click on any data point and select Source Data as shown in Figure 9.18d. Add Series 2 by clicking on Add (see Figure 9.18e). FIGURE 9.18c FIGURE 9.18d FIGURE 9.18e

5 Peterson, Technical Mathematics, 3rd edition 5 Place the cursor in the X-Values dialogue box and then move it up and click and drag along row 1, highlighting the cells that contain the years 1975 through 1999 (see Figure 9.18f). Place the cursor in the Y-Values dialogue box and then move the cursor up to row 2 and click and drag along row 2 highlighting the cells that contain the beef consumption for the years 1975 through 1999 (see Figure 9.18g). FIGURE 9.18f FIGURE 9.18g When you click OK, the result is a scatter plot with two series on the same graph. Each series should be displayed with a different color. Right click on the any data point in the first series and insert a quadratic trendline. (See Figure 9.18h.) Right click on any data point in the second series, and insert a quadratic trendline. (See Figure 9.18i.) FIGURE 9.18h FIGURE 9.18i The regressions yield the following results { t t if 4 t < 73 b(t) t t if 73 t FIGURE 9.18j where b is million pounds of beef consumption t years after 19. The final result, with some modifications in the trendlines, is shown in Figure 9.18j.