Time (seconds) Distance (feet)

0 15

1 17

2 19

4 8.1

5 0

The table shows the distance over time for a rock that was thrown from a 15-foot bridge into the water below. Using technology, create an equation to model the situation. Round values to the nearest tenth.

A$y=-1.5 x^2+4.6 x+14.7$
B$y=-1.5 x^2+4.6 x+14.8$
C$y=-2.4 x^2+5.6 x+14.9$
D$y=-0.4 x+5.3$

Question

06-11-2026
Mathematics

Answered

Time (seconds) Distance (feet)

0 15

1 17

2 19

4 8.1

5 0

The table shows the distance over time for a rock that was thrown from a 15-foot bridge into the water below. Using technology, create an equation to model the situation. Round values to the nearest tenth.

A$y=-1.5 x^2+4.6 x+14.7$
B$y=-1.5 x^2+4.6 x+14.8$
C$y=-2.4 x^2+5.6 x+14.9$
D$y=-0.4 x+5.3$

Verified Answer :

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-06T11:22:29-05:00

The data follows a non-linear path, suggesting a quadratic relationship between time and distance.
By calculating the predicted values for each model at given intervals, we determine the residuals (differences between actual and predicted).
Model $y = -1.5x^2 + 4.6x + 14.7$ yields the lowest sum of absolute differences compared to the other options.
The best-fitting model is $\boxed{y = -1.5x^2 + 4.6x + 14.7}$.

Explanation

Data Analysis and Model Evaluation Strategy

We are given a set of time-distance data points for a rock thrown from a bridge: $(0, 15)$, $(1, 17)$, $(2, 19)$, $(4, 8.1)$, and $(5, 0)$. The objective is to identify which of the four provided mathematical models best represents the path of the rock. The data shows an initial increase in distance followed by a decrease, confirming that the path is non-linear and likely follows a quadratic model, $y = ax^2 + bx + c$. The models provided are:

$y = -1.5x^2 + 4.6x + 14.7$
$y = -1.5x^2 + 4.6x + 14.8$
$y = -2.4x^2 + 5.6x + 14.9$
$y = -0.4x + 5.3$

We will evaluate the performance of these models by checking how well they predict the actual distance at each time point.

Evaluating the Models

To determine the best model, we calculate the predicted height $y$ for each model using the given time values $x = 0, 1, 2, 4, 5$. Comparing the sum of the absolute differences (residuals) between these predictions and the observed values helps identify the model with the best overall fit.

Calculating the sum of absolute errors for each model:

Model 1 ($y = -1.5x^2 + 4.6x + 14.7$): The sum of absolute differences is $0.30 + 0.80 + 1.10 + 1.00 + 0.20 = 3.40$.
Model 2 ($y = -1.5x^2 + 4.6x + 14.8$): The sum of absolute differences is $0.20 + 0.90 + 1.00 + 1.10 + 0.30 = 3.50$.
Model 3 ($y = -2.4x^2 + 5.6x + 14.9$): The sum of absolute differences is $0.10 + 1.10 + 2.50 + 9.20 + 17.10 = 30.00$.
Model 4 ($y = -0.4x + 5.3$): The sum of absolute differences is $9.70 + 12.10 + 14.50 + 4.40 + 3.30 = 44.00$.

Conclusion and Final Model Selection

By comparing the results of the total absolute differences, it is clear that Model 1 ($y = -1.5x^2 + 4.6x + 14.7$) provides the smallest sum of differences, resulting in a total error of $3.40$. Models 3 and 4 show significantly larger deviations from the observed data, and Model 2 has a slightly higher total error than Model 1. Therefore, the equation $y = -1.5x^2 + 4.6x + 14.7$ serves as the best model for this data. The final model is $\boxed{y = -1.5x^2 + 4.6x + 14.7}$.

Examples

Mathematical modeling is essential in physics and engineering to predict the behavior of moving objects. For instance, when an architect designs a fountain, they use quadratic equations to model the arc of water jets, ensuring they land precisely in the catch basin. By analyzing data points of the water's position over time, engineers can determine the optimal launch angle and velocity needed to achieve a specific aesthetic effect, just as we identified the best-fitting equation for the rock's trajectory.