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To answer the questions based on the provided table, let's go step by step.

The key points for solving histogram questions are:
1. **Widest Bar**: The width of a class in a histogram is determined by the range of the class interval.
2. **Tallest Bar**: The height of a bar in a histogram is determined by the frequency density, which is calculated as:
   \[
   \text{Frequency Density} = \frac{\text{Frequency}}{\text{Class Width}}
   \]
   where Class Width is the difference between the upper and lower bounds of the class.

### Table:
| Volume (v litres) | Frequency |
|-------------------|-----------|
| \(50 < v \leq 54\) | 10        |
| \(54 < v \leq 56\) | 15        |
| \(56 < v \leq 59\) | 24        |
| \(59 < v \leq 67\) | 28        |
| \(67 < v \leq 72\) | 39        |

### a) Widest Class (Widest Bar)
The width of each class is the difference between the upper and lower bounds:
- For \(50 < v \leq 54\): Class width = \(54 - 50 = 4\)
- For \(54 < v \leq 56\): Class width = \(56 - 54 = 2\)
- For \(56 < v \leq 59\): Class width = \(59 - 56 = 3\)
- For \(59 < v \leq 67\): Class width = \(67 - 59 = 8\)
- For \(67 < v \leq 72\): Class width = \(72 - 67 = 5\)

**Answer**: The widest class is \(59 < v \leq 67\), with a width of 8.

### b) Tallest Bar (Largest Frequency Density)
Now let's calculate the frequency density for each class:
- For \(50 < v \leq 54\): \(\text{Frequency Density} = \frac{10}{4} = 2.5\)
- For \(54 < v \leq 56\): \(\text{Frequency Density} = \frac{15}{2} = 7.5\)
- For \(56 < v \leq 59\): \(\text{Frequency Density} = \frac{24}{3} = 8\)
- For \(59 < v \leq 67\): \(\text{Frequency Density} = \frac{28}{8} = 3.5\)
- For \(67 < v \leq 72\): \(\text{Frequency Density} = \frac{39}{5} = 7.8\)

**Answer**: The class with the tallest bar is \(56 < v \leq 59\), with a frequency density of 8.

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Would you like further clarification on any part of this solution? Here are five related questions you might find helpful:
1. How does frequency density relate to histograms?
2. Can a class with a higher frequency have a shorter bar than a lower-frequency class?
3. How would you interpret a histogram visually based on this data?
4. What does a wider bar represent in a histogram compared to a taller bar?
5. Could you explain how to find the total number of observations from the data?

**Tip**: When working with histograms, always double-check the class widths since they are essential for calculating the correct bar heights (frequency density).

Work out the class which will give the widest and tallest bar for the given data in a histogram.

Analyze the given data to determine the widest class (with the largest width) and the tallest class (with the highest frequency density) when plotted on a histogram. Understand the concepts of class width and frequency density to answer this math problem, suitable for Grades 9-12.

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