https://file.aiquickdraw.com/mathbot/file/1728824445815toh3yoze.jpg

Let's break down the given problem using the histogram in the image:

### Part (a)
We need to estimate the number of rings that contain between **1.4g** and **3g** of gold.

1. **Identify relevant intervals on the x-axis (Mass in g):**
   - The mass range of interest is between 1.4g and 3g.
   - The histogram bars cover mass intervals as follows:
     - From 1g to 2g
     - From 2g to 3g
     - From 3g to 4g (but we stop at 3g)

2. **Use the frequency density to find the number of rings:**
   - The area of each bar in the histogram represents the number of rings in that interval.
   - For a histogram, the area of each bar is calculated as:
     \[
     \text{Area} = \text{Width of interval} \times \text{Frequency density}
     \]
   - We need to calculate the area for the bars between 1.4g and 3g.

#### Step-by-step calculation:
- **From 1.4g to 2g:** 
  - The height of the bar from 1g to 2g is approximately 50 (from the graph).
  - The width from 1.4g to 2g is 0.6 (since 2g - 1.4g = 0.6).
  - Area = \( 0.6 \times 50 = 30 \)

- **From 2g to 3g:**
  - The height of the bar from 2g to 3g is about 25.
  - The width from 2g to 3g is 1.
  - Area = \( 1 \times 25 = 25 \)

#### Total estimate:
\[
30 + 25 = 55 \text{ rings (approximately)}
\]

### Part (b)
The value calculated in part (a) is only an estimate because the histogram gives a **frequency density**, not exact counts. The exact boundaries of the intervals and the continuous nature of mass data also contribute to estimation rather than precise values.

Would you like further clarifications or details?

### 5 Related Questions:
1. How do you calculate the area under a histogram when intervals are unequal?
2. What is the difference between frequency density and frequency?
3. Why is it important to consider the width of the intervals in a histogram?
4. How can you improve the precision of an estimate from a histogram?
5. What would happen if the histogram bars had different widths?

### Tip:
Always multiply the **width of the interval** by the **frequency density** when interpreting histograms.

Work out an estimate for the number of rings that contain between 1.4g and 3g of gold, based on the histogram provided.

Learn how to estimate the number of rings with gold masses between 1.4g and 3g using histograms. This problem involves calculating areas under frequency density bars and interpreting histograms. Suitable for students in Grades 9-10.

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