Question

Which type of function best models the data in the table?
Time and Distance Data
Time (sec) Distance (miles)
1 3
2 9
3 27
4 81
5 243
A: quadratic B: linear C: cubic

Answer 1

To determine which type of function best models the given data, we can analyze the relationship between time (in seconds) and distance (in miles).

Let’s observe the distances provided for each second:

- At 1 sec: 3 miles
- At 2 sec: 9 miles
- At 3 sec: 27 miles
- At 4 sec: 81 miles
- At 5 sec: 243 miles

Looking at the distance values, we can see that they appear to be increasing rapidly and can be expressed as powers of 3:

- \(3 = 3^1\)
- \(9 = 3^2\)
- \(27 = 3^3\)
- \(81 = 3^4\)
- \(243 = 3^5\)

This indicates that the distance data can be expressed as:
\[
\text{Distance} = 3^t
\]
where \(t\) is the time in seconds.

This is an exponential function rather than a quadratic, linear, or cubic function. However, among the options provided:
- A: quadratic
- B: linear
- C: cubic

None of these options fit the best as the data is exponential. If you must choose from the options given, consider that if you had to approximate it using polynomials, a cubic function might be the closest in representing the overall trend, but it won't accurately represent the rapid growth seen in the data.

To summarize:
The best function appears to be exponential, but from the given choices, neither accurately represents the data. If limited to those three options, C: cubic would be the closest approximation, though it’s not a perfect match.