1. **State the problem:** We want to estimate the number of paintings over 70 years old using the histogram data. 2. **Recall the formula for frequency:** Frequency = Frequency density \( \times \) Class width. 3. **Identify relevant age intervals:** Over 70 years old means ages from 70 to 140 years. 4. **Break down the intervals and their frequency densities:** - 60 to 80 years: height about 2, width = 20 - 80 to 100 years: height about 0.5, width = 20 - 100 to 120 years: height 0, width = 20 - 120 to 140 years: height about 0.2, width = 20 5. **Calculate frequencies for each relevant part:** - For 70 to 80 (part of 60 to 80): width = 10, frequency density = 2 Frequency = 2 \( \times \) 10 = 20 - For 80 to 100: frequency = 0.5 \( \times \) 20 = 10 - For 100 to 120: frequency = 0 \( \times \) 20 = 0 - For 120 to 140: frequency = 0.2 \( \times \) 20 = 4 6. **Sum frequencies over 70 years:** $$20 + 10 + 0 + 4 = 34$$ 7. **Check given info:** 12 paintings are over 90 years old. Let's verify the frequency over 90 years: - 90 to 100 (part of 80 to 100): width = 10, frequency density = 0.5 Frequency = 0.5 \( \times \) 10 = 5 - 100 to 120: 0 - 120 to 140: 4 Total over 90 = 5 + 0 + 4 = 9 (close to 12, slight estimation difference) 8. **Final estimate:** Approximately 34 paintings are over 70 years old. This method uses the histogram's frequency density and class widths to estimate frequencies for age ranges, then sums them for the desired range.