The problem shows a quadratic function representing the height of an object in the air:

$h(t) = -4.87t^2 + 18.75t$

where $h(t)$ is the height in meters, and $t$ is the time in seconds. The task is to find the domain and range of this function, rounding values to the nearest hundredth.

1. Domain:

The domain is the set of time values ($t$) for which the height is defined. Since the object is projected into the air, the object will start at $t = 0$ and will land back on the ground when $h(t) = 0$.

• To find when the object hits the ground, set $h(t) = 0$ and solve for $t$:

$0 = -4.87t^2 + 18.75t$

Factor the quadratic equation:

$0 = t(-4.87t + 18.75)$

This gives two solutions:

• $t = 0$ (when the object is launched),
• $t = \frac{18.75}{4.87} \approx 3.85$ seconds (when the object hits the ground).

Thus, the domain is $[0, 3.85]$.

2. Range:

The range is the set of height values ($h$) that the object reaches. The maximum height occurs at the vertex of the parabola. To find the time at which the maximum height occurs, use the formula for the vertex:

$t_{\text{vertex}} = \frac{-b}{2a} = \frac{-18.75}{2(-4.87)} \approx 1.92$

Substitute $t = 1.92$ into the function $h(t)$ to find the maximum height:

$h(1.92) = -4.87(1.92)^2 + 18.75(1.92) \approx 18.05 \, \text{meters}$

Thus, the range is $[0, 18.05]$, since the height starts at 0 meters and reaches a maximum of 18.05 meters.

Final Answer:

• Domain: $[0, 3.85]$ seconds
• Range: $[0, 18.05]$ meters

Let me know if you'd like more details or explanations.

Questions to expand:

1. How is the vertex of a quadratic function determined?
2. Why is the height 0 when the object hits the ground?
3. What does the coefficient of $t^2$ tell us about the direction of the parabola?
4. How would the problem change if the initial velocity (coefficient of $t$) were different?
5. Can you apply a similar method to model other types of motion (e.g., objects dropped from a height)?

Tip: In quadratic equations modeling physical scenarios, the vertex often represents a maximum or minimum in the context of height or distance.