In this problem, we are given a quadratic equation to model the height $h(t)$ of a falling object. The equation has the form:

$h(t) = -4.9t^2 + h_0$

Here, $h(t)$ represents the height in meters after $t$ seconds, and $h_0$ is the initial height from which the object falls.

We are also provided with a table showing the time $t$ (in seconds) and the corresponding height $h$ (in meters) for a pebble falling from a cliff:

The goal is to select the correct quadratic equation based on the initial height $h_0$.

Step 1: Determine $h_0$

We are given the height at $t = 1$ second, which is 55.1 meters. If we substitute $t = 1$ into the general equation, we can find $h_0$:

$h(1) = -4.9(1)^2 + h_0 = 55.1$

Solving for $h_0$:

$-4.9 + h_0 = 55.1$ $h_0 = 55.1 + 4.9 = 60$

Thus, the initial height $h_0$ is 60 meters.

Step 2: Select the correct equation

The correct equation that models the situation is:

$h(t) = -4.9t^2 + 60$

In the dropdown, this corresponds to the option: $h(t) = -4.9t^2 + 60$

This is the correct choice because it matches both the form of the equation and the initial height we calculated.

Would you like further details or have any other questions?

Here are some related questions for you:

1. How is the value of $-4.9$ derived in the height formula for free-falling objects?
2. What is the significance of $h_0$ in this type of equation?
3. How can you verify if this equation holds true for the other values of $t$ in the table?
4. Can you describe how gravity affects the object's height over time using this equation?
5. How would the equation change if the pebble were thrown upwards instead of falling?

Tip: In free-fall problems, the coefficient $-4.9$ comes from half the gravitational acceleration on Earth, which is approximately $9.8 \, \text{m/s}^2$.