- The problem provides a quadratic model \(h(t) = -4.9t^2 + h_0\) and data points for the height at specific times.
- By substituting each data point \((t, h)\) into the model, we solve for the initial height \(h_0\) in each case:
- For \(t=1, h=55.1 \Rightarrow h_0 = 60\).
- For \(t=2, h=40.4 \Rightarrow h_0 = 60\).
- For \(t=3, h=15.9 \Rightarrow h_0 = 60\).
- Since \(h_0 = 60\) is consistent across all points, the model is \(h(t) = -4.9t^2 + 60\).
- The correct choice is \(\boxed{h(t) = -4.9t^2 + 60}\).
Explanation
- Understanding the Problem and Data
We are given the quadratic function \(h(t) = -4.9t^2 + h_0\), which models the height of an object falling from an initial height \(h_0\). We are also provided with a table containing data for a pebble falling from a cliff at different times:
- For \(t=1 \text{ s}\), the height is \(h=55.1 \text{ m}\).
- For \(t=2 \text{ s}\), the height is \(h=40.4 \text{ m}\).
- For \(t=3 \text{ s}\), the height is \(h=15.9 \text{ m}\).
Our objective is to find the value of \(h_0\) that is consistent across all data points and determine the correct quadratic equation.
- Calculating the Initial Height h_0
To find the constant \(h_0\), we substitute each given time \(t\) and its corresponding height \(h\) into the model equation \(h(t) = -4.9t^2 + h_0\).
For \(t=1\):
\[55.1 = -4.9(1)^2 + h_0\]
\[55.1 = -4.9 + h_0\]
\[h_0 = 55.1 + 4.9 = 60\]
For \(t=2\):
\[40.4 = -4.9(2)^2 + h_0\]
\[40.4 = -4.9(4) + h_0\]
\[40.4 = -19.6 + h_0\]
\[h_0 = 40.4 + 19.6 = 60\]
For \(t=3\):
\[15.9 = -4.9(3)^2 + h_0\]
\[15.9 = -4.9(9) + h_0\]
\[15.9 = -44.1 + h_0\]
\[h_0 = 15.9 + 44.1 = 60\]
In all cases, the initial height \(h_0\) is consistent at \(60\).
- Determining the Final Model
Based on our calculation that \(h_0 = 60\), the quadratic equation that models the situation is:
\[h(t) = -4.9t^2 + 60\]
Comparing this result with the given options:
A\(h(t) = -4.9t^2 + 66\)
B\(h(t) = -4.9t^2 + 60\)
C\(h(t) = -4.9t^2 + 50.2\)
D\(h(t) = -4.9t^2 + 55.1\)
We can see that the correct option is (b). The smallest possible value of the initial height consistent with the provided data is \(\boxed{h(t) = -4.9t^2 + 60}\).
Examples
This problem demonstrates how scientists and engineers use data points to model physical phenomena like gravity. By observing an object in motion and measuring its position over time, one can determine the parameters of a mathematical model, such as the initial height or acceleration constant. This is essential in fields like ballistics, aviation, and structural engineering to predict future behavior or verify theoretical models in real-world conditions.