SECTION 3.4 Building Quadratic Models from Verbal Descriptions and from Data 175 Figure 32 (b) Use a graphing utility of find the quadratic function of best fit that models the relation between temperature and efficiency. (c) Use the model found in part (b) to approximate the temperature at which efficiency is greatest. (d) Use the model found in part (b) to approximate the highest efficiency. (e) Use a graphing utility to draw the quadratic function of best fit on the scatter plot. Figure 31 Solution (a) Enter the data into a table using Desmos. Figure 31 shows the scatter plot using Desmos. It appears the data follow a quadratic relation that is concave down. That is, a 0. < (b) Enter y ax bx c ~ 1 1 2 1 + + into Desmos to obtain the results shown in Figure 32. The quadratic function of best fit that models that relation between temperature and efficiency is E x x x 0.2767 7.3328 45.3253 2 ( ) = − + + The Model where x represents the temperature and E represents the efficiency (in percent). (c) Based on the quadratic function of best fit, the temperature with the greatest efficiency is x b a2 7.3328 2 0.2767 13.3 degrees Celsius ( ) = − = − − ≈ (d) Evaluate the function E x( ) at x 13.3. = E 13.3 0.2767 13.3 7.3328 13.3 45.3253 93.9 2 ( ) = − ⋅ + ⋅ + ≈ According to the model, a temperature of 13.3 degrees Celsius has the greatest efficiency of 93.9 percent. (e) Figure 33 shows the graph of the quadratic function of best fit found in part (b) drawn on the scatter plot. Now Work PROBLEM 17 Figure 33 E x x x 0.2767 7.3328 45.3253 2 ( ) =− + +
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