Question

A hot air balloon descends to the ground. The function h(t) = 210 – 15t can be used to describe the altitude of the balloon as it approaches the ground. Which statement best describes the graph of the function that models the descent of the balloon?

The graph is discrete because there cannot be fractional values for time.
The graph is discrete because there cannot be negative values for altitude.
The graph is continuous because there can be fractional values for time.
The graph is continuous because there can be negative values for altitude.

Answer 1

[AI-generated answer — Bot GPT-4o]

The function \( h(t) = 210 - 15t \) describes the altitude \( h \) of the hot air balloon as a function of time \( t \). Let's analyze whether the graph of this function is discrete or continuous based on the nature of the variables involved.

1. **Discrete vs Continuous Variables:**
- A graph is **discrete** if it consists of isolated points. Discrete data are countable and typically involve integers (like the number of people).
- A graph is **continuous** if it includes an unbroken set of points. Continuous data can take on any value within a range (like height, weight, time).

2. **Behavior of the variables:**
- In this context, time \( t \) is considered to be a continuous variable because time can be measured in infinitely small units, i.e., there can be fractional values for time.
- As long as the time is within a realistic range (assuming it's positive or zero), the altitude \( h \) will also be a continuous function decreasing linearly from 210 units as time progresses.

Given the continuous nature of time and the form of the function (a linear equation), the graph of the function \( h(t) = 210 - 15t \) will be a straight line, representing a continuous descent.

Therefore, the correct statement is:
- **The graph is continuous because there can be fractional values for time.**