Question

1. a set of data was used to create a linear, quadratic, and an exponential regression model. the residual plots for the three models are shown above. based on the three residual plots, which of the following could be an appropriate model for the data? (a) y = 1.74x - 3.82 (b) y = 0.25x² - 2.31x + 1.87 (c) y = 0.4(1.82)^x (d) y = 1.3logx - 2.1 19. at a water - treatment plant, a pump begins to process wastewater at time t = 0 minutes. the first pump processes water at a constant rate of 42 gallons per minute. three minutes later, a second pump is brought online and pumps at a constant rate of 50 gallons per minute so that both pumps are simultaneously processing wastewater. both pumps continue to process water at their respective rates until t = 30 minutes. let g represent the total number of gallons of water processed by the two pumps after t minutes. which of the following piece - wise functions is an appropriate model for the function g? (a) g(t) = {42t, if 0 ≤ t ≤ 3; 50t, if 3 < t ≤ 30} (b) g(t) = {42t, if 0 ≤ t ≤ 3; 50(t - 3), if 3 < t ≤ 30} (c) g(t) = {42t, if 0 ≤ t ≤ 3; 50t + 126, if 3 < t ≤ 30} (d) g(t) = {42t, if 0 ≤ t ≤ 3; 50(t - 3)+42t, if 3 < t ≤ 30}

Explanation:

18.

Step1: Analyze residual plots

In a good - fitting regression model, the residual plot should have no discernible pattern. The linear regression model residual plot has a random scatter of points around the horizontal axis, indicating that a linear model may be appropriate. The quadratic and exponential regression model residual plots show some patterns, suggesting these models may not fit the data well.

Step2: Identify linear model

The equation \(y = 1.74x-3.82\) is a linear model of the form \(y=mx + b\).

Step1: Analyze the first 3 minutes

For \(0\leq t\leq3\), only the first pump is working, and it processes water at a rate of 42 gallons per minute. So the amount of water processed \(G(t)=42t\).

Step2: Analyze after 3 minutes

For \(t > 3\), the first pump has been working for \(t\) minutes and the second pump has been working for \((t - 3)\) minutes. The first pump processes water at a rate of 42 gallons per minute and the second pump at a rate of 50 gallons per minute. The amount of water processed by the first pump in \(t\) minutes is \(42t\), and the amount of water processed by the second pump in \((t - 3)\) minutes is \(50(t - 3)\). So \(G(t)=50(t - 3)+42t\).

Answer:

A. \(y = 1.74x-3.82\)

19.