Question

The table shows the heights h (in feet) of a sponge t seconds after it was dropped by a window cleaner on top of a skyscraper. $$ \begin{array}{|l|c|c|c|c|c|} \hline \text { Time, } \boldsymbol{t} & 0 & 1 & 1.5 & 2.5 & 3 \\ \hline \text { Height, } \boldsymbol{h} & 280 & 264 & 244 & 180 & 136 \\ \hline \end{array} $$ a. Use a graphing calculator to create a scatter plot. Which better represents the data, a line or a parabola? Explain. b. Use the regression feature of your calculator to find the model that best fits the data. c. Use the model in part (b) to predict when the sponge will hit the ground. d. Identify and interpret the domain and range in this situation.

Short answer

The data is better represented as a parabola. The quadratic regression model will be used to calculate the best fit. Using this model, the time when the sponge hits the ground is calculated by setting \(h=0\), and solving for 't'. The domain represents the time from when the sponge is dropped until it hits the ground and the range is the possible heights from the skyscraper to the ground.

Step-by-step solution

Create a Scatter Plot

Using a graphing calculator, plot the points (0,280), (1, 264), (1.5, 244), (2.5, 180), and (3,136). Each point corresponds to the time and the height of the sponge. Analyze if the data is represented better as a line or a parabola. In this case, since the object is moving downwards and since it's falling because of the gravity, the shape would look like a part of a parabola.

Determine the Best Fit

Use the regression feature of the graphing calculator to find the quadratic regression model, which is the best fit for the data. The quadratic regression model of this form: \( h(t) = at^2 + bt + c \) will be given by the calculator.

Predict the Sponge's Impact Time

Use the regression model \( h = at^2 + bt + c \) obtained from the calculator to predict when the sponge will hit the ground. It's when h=0. Solve the equation for 't' to get this time.

Identify and Interpret Domain and Range

In this context, the domain represents the time starting from when the sponge is dropped to when it hits the ground. The range represents the possible heights of the sponge, from the height of the skyscraper to the ground level. Domain is [0, T] where T is when the sponge hits the ground and Range is [0, 280] in feet.

Key concepts

Understanding Scatter Plots

A scatter plot is a graph used to display and analyze the relationship between two sets of data to see if there's a correlation or trend. When creating a scatter plot, each point on the graph represents an individual data pair, with one variable on the x-axis and the other on the y-axis.

For instance, when we examine the heights of a falling sponge from a skyscraper at different time intervals, we plot time on the x-axis and height on the y-axis. The pattern that the points create can suggest the type of relationship—linear, quadratic, or none—that might exist between the time and height. In the sponge's case, the trajectory of the fall will likely resemble a curve, implying a quadratic relationship. This is because the sponge's downward motion under gravity follows a parabolic path, not a straight line. Consequently, a quadratic equation, which graphs as a parabola, often better represents such physical data.

Domain and Range in Real-World Contexts

The domain of a function is essentially the set of all possible input values, while the range is the set of all possible output values. To understanding real-world problems through a mathematical lens, identifying these two is crucial.

In our falling sponge scenario, the domain is the time during which the sponge's fall is being recorded. It's set from the moment the sponge is dropped (at time zero) until it hits the ground. This is a practical application where the domain has clear physical constraints. The range, on the other hand, represents the sponge's possible heights, starting from the skyscraper's top to the ground. Recognizing the domain and range helps to create an accurate graphical representation and can be influential in making predictions based on the model.

Predicting Outcomes with Regression

When we talk about predicting outcomes with regression, we're generally referring to using the best-fit equation derived from a set of data to estimate or predict unknown values. Regression analysis helps us understand how the typical value of the dependent variable changes when any one of the independent variables is varied.

In the example of the falling sponge, by performing quadratic regression, we derive an equation that models the sponge's height over time with a parabolic equation. Using this model, we can predict when the sponge will hit the ground by setting the height to zero and solving the quadratic equation for the time 't'. Such an application is not only useful for academic exercises but also for real-world applications where predicting future events based on past data is vital.