1. **Problem statement:** We are given a histogram showing the frequency density of Roman coins by their mass intervals. We know that 108 coins weigh between 8 g and 17 g. We need to find the total number of coins in the museum's collection. 2. **Understanding the histogram:** The histogram bars represent frequency density on the y-axis and mass intervals on the x-axis. The area of each bar (width × height) gives the frequency (number of coins) in that mass interval. 3. **Given data:** - Interval 0–5 g: height = 3 - Interval 5–10 g: height = 7 - Interval 10–17 g: height = 2 - Interval 17–22 g: height = 4 4. **Calculate the number of coins between 8 g and 17 g:** - The interval 8–10 g is part of the 5–10 g bar (height 7). - The interval 10–17 g corresponds to the 10–17 g bar (height 2). 5. **Calculate frequency for 8–10 g:** - Width = $10 - 8 = 2$ - Height = 7 - Frequency = width × height = $2 \times 7 = 14$ 6. **Calculate frequency for 10–17 g:** - Width = $17 - 10 = 7$ - Height = 2 - Frequency = $7 \times 2 = 14$ 7. **Total frequency for 8–17 g:** - $14 + 14 = 28$ 8. **Given that 108 coins weigh between 8 g and 17 g, but our calculation shows 28, this means the frequency density scale is off by a factor.** 9. **Find the scale factor:** - Scale factor = $\frac{108}{28} = 3.8571$ 10. **Calculate total number of coins:** - Calculate frequency for each interval using the scale factor. - Interval 0–5 g: - Width = 5 - Height = 3 - Frequency = $5 \times 3 = 15$ - Adjusted frequency = $15 \times 3.8571 = 57.86$ - Interval 5–10 g: - Width = 5 - Height = 7 - Frequency = $5 \times 7 = 35$ - Adjusted frequency = $35 \times 3.8571 = 135$ - Interval 10–17 g: - Width = 7 - Height = 2 - Frequency = $7 \times 2 = 14$ - Adjusted frequency = $14 \times 3.8571 = 54$ - Interval 17–22 g: - Width = 5 - Height = 4 - Frequency = $5 \times 4 = 20$ - Adjusted frequency = $20 \times 3.8571 = 77.14$ 11. **Sum all adjusted frequencies:** - $57.86 + 135 + 54 + 77.14 = 324$ 12. **Final answer:** - The total number of Roman coins in the museum's collection is approximately **324**.