1. The problem states that 108 coins weigh between 8 g and 17 g. We need to find the total number of Roman coins in the museum's collection. 2. The histogram shows frequency density for mass intervals: - 0 to 5 g: height 1 - 5 to 10 g: height 4 - 10 to 15 g: height 1 - 15 to 20 g: height 2 - 20 to 25 g: height 0 3. Frequency density means frequency = height \times width of the interval. 4. Calculate frequencies for each interval: - 0 to 5 g: $1 \times 5 = 5$ - 5 to 10 g: $4 \times 5 = 20$ - 10 to 15 g: $1 \times 5 = 5$ - 15 to 20 g: $2 \times 5 = 10$ - 20 to 25 g: $0 \times 5 = 0$ 5. The coins weighing between 8 g and 17 g span parts of the 5 to 10 g, 10 to 15 g, and 15 to 20 g intervals. 6. Find the fraction of each relevant interval that lies between 8 g and 17 g: - 5 to 10 g interval (5 g width): from 8 to 10 g is 2 g, fraction = $\frac{2}{5}$ - 10 to 15 g interval (5 g width): fully included, fraction = 1 - 15 to 20 g interval (5 g width): from 15 to 17 g is 2 g, fraction = $\frac{2}{5}$ 7. Calculate the number of coins in each part: - 5 to 10 g: $20 \times \frac{2}{5} = 8$ - 10 to 15 g: $5 \times 1 = 5$ - 15 to 20 g: $10 \times \frac{2}{5} = 4$ 8. Total coins between 8 g and 17 g according to histogram frequencies: $8 + 5 + 4 = 17$ 9. Given that these 17 coins correspond to 108 actual coins, find the scale factor: $$\text{Scale factor} = \frac{108}{17} \approx 6.3529$$ 10. Calculate total number of coins by summing all frequencies and multiplying by scale factor: - Total frequency = $5 + 20 + 5 + 10 + 0 = 40$ - Total coins = $40 \times 6.3529 \approx 254$ **Final answer:** There are approximately **254 Roman coins** in the museum's collection.