To analyze the function [tex]\( h(t) = 210 - 15t \)[/tex] and the nature of its graph, let's examine the components and characteristics of the function in detail.
1. Understanding the Function:
- The function [tex]\( h(t) = 210 - 15t \)[/tex] describes the altitude [tex]\( h \)[/tex] of the balloon at any given time [tex]\( t \)[/tex].
- [tex]\( h \)[/tex] is the altitude in meters, and [tex]\( t \)[/tex] is the time in minutes.
2. Nature of the Variables:
- Time, [tex]\( t \)[/tex]: Time is typically a continuous variable, as it can take any positive real value, including fractional values (e.g., 1.5 minutes).
- Altitude, [tex]\( h(t) \)[/tex]: As the balloon descends, the altitude decreases. Since the function is linear and continuous, the altitude [tex]\( h(t) \)[/tex] can also take on any real value within its range, including fractional values.
3. Graph of the Function:
- The graph of a linear function such as [tex]\( h(t) = 210 - 15t \)[/tex] is a straight line.
- In this context, we consider the practical aspects:
- Fractional Values for Time: Time can indeed have fractional values (e.g., 1.5 minutes, 2.75 minutes), implying that the function can take on corresponding fractional values for altitude.
- Fractional Values for Altitude: Since the function is continuous, altitude can also be represented with fractional values.
4. Continuous vs. Discrete:
- A continuous graph is one where any point along the line can exist, meaning there are no breaks, jumps, or holes in the graph.
- A discrete graph involves distinct, separate points, often representing countable values.
Given that both time and altitude can take on fractional values, the proper characterization of the function and its graph is based on this continuity.
5. Evaluating the Statements:
- "The graph is discrete because there cannot be fractional values for time" is incorrect because time can indeed be fractional.
- "The graph is discrete because there cannot be negative values for altitude" is incorrect; this statement confuses negativity with discreteness and doesn't address fractional values.
- "The graph is continuous because there can be fractional values for time" is correct. This statement properly recognizes the continuity of the function.
- "The graph is continuous because there can be negative values for altitude" is incorrect within the practical context, as altitude typically remains non-negative.
Therefore, the best description of the graph of the function [tex]\( h(t) = 210 - 15t \)[/tex] is:
"The graph is continuous because there can be fractional values for time."
