1. **State the problem:** We are given a histogram showing the frequency density of Roman coins by mass intervals. We know that 126 coins weigh between 8 g and 17 g, and we need to find the total number of coins in the collection. 2. **Recall the formula for frequency:** $$\text{Frequency} = \text{Frequency density} \times \text{Class width}$$ This means the number of coins in each mass interval equals the height of the bar (frequency density) multiplied by the width of the interval. 3. **Calculate the frequency for the interval 8 g to 17 g:** - Frequency density = 2 - Class width = 17 - 8 = 9 So, $$\text{Frequency} = 2 \times 9 = 18$$ But the problem states there are actually 126 coins in this interval, so the histogram must be scaled by a factor to match this. 4. **Find the scale factor:** $$\text{Scale factor} = \frac{126}{18} = 7$$ This means all frequencies from the histogram should be multiplied by 7 to get the actual number of coins. 5. **Calculate frequencies for all intervals using the scale factor:** - Interval 0 to 5: - Frequency density = 1 - Width = 5 - Frequency = 1 \times 5 = 5 - Actual coins = 5 \times 7 = 35 - Interval 5 to 8: - Frequency density = 6 - Width = 3 - Frequency = 6 \times 3 = 18 - Actual coins = 18 \times 7 = 126 - Interval 8 to 17: - Already given as 126 coins - Interval 17 to 22: - Frequency density = 3 - Width = 5 - Frequency = 3 \times 5 = 15 - Actual coins = 15 \times 7 = 105 - Interval 22 to 25: - Frequency density = 0 - Width = 3 - Frequency = 0 \times 3 = 0 - Actual coins = 0 6. **Find total number of coins:** $$\text{Total} = 35 + 126 + 126 + 105 + 0 = 392$$ **Final answer:** There are 392 Roman coins in the museum's collection in total.