145 1.5 Applications and Modeling with Quadratic Equations Modeling with Quadratic Equations EXAMPLE 4 Analyzing Trolley Ridership The I-Ride Trolley service carries passengers along the International Drive resort area of Orlando, Florida. The line graph in Figure 8 shows I-Ride Trolley ridership data in millions. The quadratic equation y = -0.00645x2 + 0.103x + 1.62 models ridership from 2000 to 2018, where y represents ridership in millions, and x = 0 represents 2000, x = 1 represents 2001, and so on. (a) Use the model to estimate ridership in 2018 to the nearest tenth of a million. Compare the result to the actual ridership figure of 1.3 million. (b) According to the model, in what year did ridership reach 1.8 million? SOLUTION (a) Because x = 0 represents the year 2000, use x = 18 to represent 2018. y = -0.00645x2 + 0.103x + 1.62 Given model y = -0.0064511822 + 0.1031182 + 1.62 Let x = 18. y ≈1.4 million Use a calculator. The prediction is about 0.1 million (that is, 100,000) greater than the actual figure of 1.3 million. (b) y = -0.00645x2 + 0.103x + 1.62 Given model 1.8 = -0.00645x2 + 0.103x + 1.62 Let y = 1.8. 0 = -0.00645x2 + 0.103x - 0.18 Standard form x = -0.103{210.10322 - 41-0.0064521-0.182 21-0.006452 Quadratic formula x ≈2.0 or x ≈14.0 Use a calculator. The year 2002 corresponds to x = 2.0. Thus, according to the model, ridership reached 1.8 million in the year 2002. This outcome seems reasonable compared to the line graph. The year 2014 corresponds to x = 14.0 and seems reasonable compared with the line graph value. Because the number of riders on the I-Ride Trolley service increases until about 2008 and then decreases after that, there are two years during which ridership is approximately 1.8 million. Notice that both years 2002 and 2014 are within the scope of the data used. If either result had fallen beyond the data used in the model, we would view that result with skepticism because the model might not continue as given. S Now Try Exercise 49. 0 0.5 1.0 1.5 2.5 2.0 Millions ’00 ’05 ’10 Year Data from I-Ride Trolley, International Drive Business Improvement District, www.idrivedistrict.com Ridership ’15 ’20 Figure 8 Solve this equation for x. LOOKING AHEAD TO CALCULUS In calculus, you will need to be able to write an algebraic expression from the description in a problem like those in this section. Using calculus techniques, you will be asked to find the value of the variable that produces an optimum (a maximum or minimum) value of the expression.
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