To choose the quadratic equation that models the situation, we need to start with the given function that describes the height [tex]\( h(t) \)[/tex] at any time [tex]\( t \)[/tex]:
[tex]\[ h(t) = -4.9t^2 + h_0 \][/tex]
Our goal is to determine the initial height [tex]\( h_0 \)[/tex] using the data provided in the table. The data points given are:
- At [tex]\( t = 1 \)[/tex] second, [tex]\( h = 55.1 \)[/tex] meters
- At [tex]\( t = 2 \)[/tex] seconds, [tex]\( h = 40.4 \)[/tex] meters
- At [tex]\( t = 3 \)[/tex] seconds, [tex]\( h = 15.9 \)[/tex] meters
Let's use these data points to find [tex]\( h_0 \)[/tex].
### Step-by-Step Solution
1. Substitute the first data point into the equation:
[tex]\[ h(1) = -4.9(1)^2 + h_0 = 55.1 \][/tex]
[tex]\[ -4.9 + h_0 = 55.1 \][/tex]
[tex]\[ h_0 = 55.1 + 4.9 \][/tex]
[tex]\[ h_0 = 60.0 \][/tex]
2. Substitute the second data point into the equation:
[tex]\[ h(2) = -4.9(2)^2 + h_0 = 40.4 \][/tex]
[tex]\[ -4.9 \cdot 4 + h_0 = 40.4 \][/tex]
[tex]\[ -19.6 + h_0 = 40.4 \][/tex]
[tex]\[ h_0 = 40.4 + 19.6 \][/tex]
[tex]\[ h_0 = 60.0 \][/tex]
3. Substitute the third data point into the equation:
[tex]\[ h(3) = -4.9(3)^2 + h_0 = 15.9 \][/tex]
[tex]\[ -4.9 \cdot 9 + h_0 = 15.9 \][/tex]
[tex]\[ -44.1 + h_0 = 15.9 \][/tex]
[tex]\[ h_0 = 15.9 + 44.1 \][/tex]
[tex]\[ h_0 = 60.0 \][/tex]
### Conclusion
From all three data points, we consistently find that the initial height [tex]\( h_0 \)[/tex] is [tex]\( 60.0 \)[/tex] meters. Therefore, the quadratic equation that models the situation is:
[tex]\[ h(t) = -4.9t^2 + 60.0 \][/tex]
