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The problem asks to determine the values of **A** and **B** when plotting time (seconds) versus height (meters) on a graph, with the base-10 logarithmic scale being used for the horizontal axis (time).

### Analyzing the Data:

The time values provided are:
\[
\text{Time (seconds)} = 81, 165, 17, 7, 27
\]

The heights corresponding to those times are:
\[
\text{Height (m)} = 80, 67, 80, 21, 57
\]

The horizontal axis uses a logarithmic scale, and we're asked for the values of **A** and **B**, which likely represent the limits of the scale on the time axis.

### Determining **A** and **B**:

- **A** is likely the lower bound on the time axis (logarithmic scale), corresponding to the smallest time value.
- **B** is likely the upper bound on the time axis (logarithmic scale), corresponding to the largest time value.

The smallest time value is \( t_{\min} = 7 \) seconds, and the largest time value is \( t_{\max} = 165 \) seconds.

### Logarithmic Scale:
In a base-10 logarithmic scale, the value at any point on the axis is represented by the logarithm of the actual value.

Thus:
\[
A = \log_{10}(7) \quad \text{and} \quad B = \log_{10}(165)
\]

I'll calculate these values now.The values of **A** and **B** are:

- \( A \approx 0.85 \) (the logarithmic value of the smallest time, 7 seconds)
- \( B \approx 2.22 \) (the logarithmic value of the largest time, 165 seconds)

Would you like more details or have any further questions?

Here are some follow-up questions to deepen your understanding:
1. How would the graph look if a linear scale were used instead of a logarithmic one?
2. Why is the logarithmic scale useful when plotting data with large differences in values?
3. What impact would changing the base of the logarithmic scale have on the values of \( A \) and \( B \)?
4. How could you convert a logarithmic scale back to a linear one?
5. How do you determine the appropriate range for a logarithmic scale?

**Tip**: When plotting data on a logarithmic scale, ensure that all data points are positive, as logarithms of negative numbers or zero are undefined.

A graph is being plotted for time (seconds) vs. height (meters). Time is plotted on the horizontal axis using a base-10 logarithmic scale. What are the values of A and B?

This problem explores how to plot time vs. height using a base-10 logarithmic scale on the horizontal axis. Given the time values ranging from 7 to 165 seconds, we calculate the logarithmic values to find the scale limits A and B.

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