Students in a lab measured the breaking strength S (in pounds) of wood 2 inches thick, x inches high, and 12 inches long. The results are shown in the table. X 4 6 8 10 12 S 2,802 6,314 11,176 17,888 25,761 (a) Use the regression capabilities of a graphing utility to fit a quadratic model to the data. (Round your coefficients to three decimal places.) S ( x ) = (b) Use a graphing utility to plot the data and graph the model. S 30 000 500 25 000 480 20 000 460 15 000 440 10 000 420 5000 400 380 2 X O 4 6 8 10 12 2 X O 4 6 8 10 12 S 25 000 25 000 20 000 20 000 15 000 15 000 10 000 10 000 5000 5000 X O 2 8 10 12 X 2 6 8 10 12 (c) Use the model to approximate the breaking strength when x = 2. (Round your answer to three decimal places.) IbA V8 car engine is coupled to a dynamometer, and the horsepower y is measured at different engine speeds x (in thousands of revolutions per minute). The results are shown in the table. x 1 2 3 4 5 6 y 90 129 173 209 238 216 (a) Use the regression capabilities of a graphing utility to find a cubic model for the data. (Round your coefficients to three decimal places.) y ( x ) = -2.565x + 20.413x2 - 7.308x + 80 X (b) Use a graphing utility to plot the data and graph the model. y 300 y 300 250 250 200 200 150 150 100 100 50 50 2 X O 1 3 4 5 6 7 O 1 2 3 X 4 5 6 7 y y 300 300 250 250 200 200 150 150 100 100 50 50 X O 1 2 3 4 5 6' 7 O 2 3 4 X 5 6(c) Use the model to approximate the horsepower when the engine is running at 3500 revolutions per minute. (Round to the nearest whole number.) -1 X horsepower Additional Materials eBookSTEP 5: Find q by observing that a peak occurs at 7/4, and a trough occurs at 37/4. Q = STEP 6: Find D by taking the average of the maximum and minimum y-values. D = ()min + ymax)/2 + ) /2 STEP 7: Give a model for the data. Use a graphing utility to graph the model. Compare the result with the data in the figure. y =The motion of an oscillating weight suspended by a spring was measured by a motion detector. The data collected and the approximate maximum (positive and negative) displacements from equilibrium are shown in the figure. The displacement y is measured in centimeters and the time t is measured in seconds. Use the form y = A sin(wt + (p) + D to find a model for the data. (0.1, 5.99) 5- 3- 21 (0.3, 4.01) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 STEP 1: Is y a function of t? Yes No STEP 2: Approximate the amplitude A by taking half the difference between the maximum and minimum y-values. A = ()max - ymin)/2 / 2 STEP 3: Approximate the period T by taking two times the time elapsed from the trough to the peak. T = 2(ttrough - tpeak) STEP 4: Find w using the period T 2TC W = 2 1(d) How many times greater is the breaking strength for a 4-inch-high board than for a 2-inch-high board? (Round y (e) How many times greater is the breaking strength for a 12-inch-high board than for a 6-inch high board? (Round your answer to two decimal places.) When the height of the board increases by a factor, does the breaking strength increase by the same factor? Explain. When the height is ---Select--- @ , the breaking strength increases approximately by a factor of --?-- 0