Let's analyze the question step by step.

The function given for the Giant Slide is: $h = -8t^2 + 56$ where $h$ represents the height and $t$ represents the distance down the slide. The question asks:

"Which set of numbers represents an appropriate domain for the function?"

Key Observations:

1. Domain: This refers to the possible values that $t$ can take. For a physical context like a slide, $t$ must represent a distance down the slide.
2. Distance cannot be negative: Since $t$ represents a physical distance, the domain must include values greater than or equal to 0.
3. End of the slide: The slide starts at $t = 0$ and ends at some maximum value of $t$, which corresponds to when the height $h$ becomes zero.

Step 1: Solve for when $h = 0$

To determine the range of $t$ (distance), set $h = 0$: $-8t^2 + 56 = 0$ Simplify: $t^2 = \frac{56}{8} = 7$ Take the square root: $t = \sqrt{7} \quad \text{or} \quad t = -\sqrt{7}$ Since $t$ represents distance, negative values are not valid. Therefore: $t \in [0, \sqrt{7}]$

Step 2: Choose the appropriate domain set

The options are:

• A. Positive integers: Includes only whole numbers like 1, 2, 3... This is not correct since $t$ can be any real number between 0 and $\sqrt{7}$.
• B. Positive real numbers: Includes all positive numbers, including decimals. This is partially correct but excludes 0.
• C. Natural numbers: Typically starts at 1, so it excludes 0 and decimals. Not correct.
• D. Rational numbers: Includes all numbers that can be expressed as fractions (like 0, 0.5, 1.2, etc.), which can approximate real numbers within the domain.

Final Answer:

The appropriate domain for $t$ includes all real numbers between 0 and $\sqrt{7}$, including 0. Therefore, the correct choice is:

B. Positive real numbers

Do you want a deeper explanation or have further questions?

Here are 5 related questions to expand understanding:

1. What is the range of the function $h = -8t^2 + 56$?
2. How would the domain change if the function represented a real-world projectile motion?
3. What is the significance of the negative sign in the function $-8t^2$?
4. What are rational numbers, and how do they differ from real numbers?
5. Can the function $h = -8t^2 + 56$ represent any physical situation other than a slide?

Tip: When interpreting real-world functions, always consider practical limitations like time, distance, and physical boundaries.