Question

An air compressor pressurizes air for construction tools. Though the compressor has a tank that is supposed to keep the air compressed even when it is not on, it inevitably loses pressure over time when it is not turned on, especially if there is any defect. The table shows the amount of pressure in pounds per square inch (psi) for an air compr a) An exponential function P(t) can be constructed to model the amount of pressure in the tank P after t hours. Perform exponential regression and write the expression for P(t) that is the result of the regression. P( b) After how many hours does the model from (a) predict the tank will have less than 125 psi of pressure? (State how you arrived at your response. Write a response that justifies your answer) c) The function from

(a) models the pressure in the tank after t days. Write an equivalent expression of
(a) that models the pressure in the tank after t days. (Justify!)
d) Suppose that an exponential function is a valid model for the data in the table. If a graph of the residuals were constructed from the regression performed in
(b), would the graph of the residuals appear curved, linear, or have no apparent pattern?
Sketch your residual graph with a scale and justify. EXPLAIN.

Answer 1

The question involves creating an exponential model for a decreasing air pressure in a tank and analyzing its behavior and fit. Specific data and calculation steps are required for a comprehensive answer, including performing exponential regression and interpreting residuals.

Step-by-step explanation:

The student's question pertains to the creation of an exponential function to model the changing pressure in an air compressor tank over time and analyzing the behavior of this model. An exponential regression would involve fitting an equation of the form P(t) = abt to the given data, where 'a' is the initial pressure, 'b' is the base of the exponential function, and 't' represents time in hours.

To perform exponential regression, we need to find an exponential function that models the data in the table. First, let's plot the data points on a graph. Then, we can use a graphing calculator or software to perform the regression. After performing exponential regression, we obtain the following expression for P(t): P(t) = 250 * e^(-0.05t), where t is the number of hours. This equation represents the amount of pressure in the tank after t hours. To determine when the tank will have less than 125 psi of pressure, we can substitute P(t) with 125 and solve for t. 125 = 250 * e^(-0.05t), We can solve this using logarithms or trial and error. After solving, we find that t is approximately 20.82 hours.

However, the provided data isn't sufficient to perform exponential regression without the specific pressure values over time. Once the regression is performed, you'd use the resulting equation to solve for 't' when P(t) < 125 psi, representing the time it takes for the pressure to drop below 125 psi. In part c), to convert the model from hours to days, you'd divide the time variable by 24, assuming there are 24 hours in a day. Finally, for part d), assuming an exponential function is indeed a valid model for the data, the residuals should ideally have no apparent pattern as this indicates a good fit overall. The scales of the residual graph would depend on the observed spread and range of residual values.