https://file.aiquickdraw.com/mathbot/file/1730021426199yfcqo66n.jpg

Let's solve each part of the question based on the given histogram.

### Part (a)
To estimate the number of rings with a mass between 1.4 g and 3 g of gold, we should look at the histogram bars within this range and calculate the frequency for each relevant bar.

- **Bar from 1 g to 2 g**:
  - Mass range: 1 g to 2 g.
  - Frequency density: 50.
  - Width of bar: \(2 - 1 = 1\).
  - Frequency (number of rings) = Frequency density \(\times\) Width = \(50 \times 1 = 50\).

- **Bar from 2 g to 2.5 g**:
  - Mass range: 2 g to 2.5 g.
  - Frequency density: 60.
  - Width of bar: \(2.5 - 2 = 0.5\).
  - Frequency (number of rings) = \(60 \times 0.5 = 30\).

- **Bar from 2.5 g to 3 g**:
  - Mass range: 2.5 g to 3 g.
  - Frequency density: 20.
  - Width of bar: \(3 - 2.5 = 0.5\).
  - Frequency (number of rings) = \(20 \times 0.5 = 10\).

Adding up the frequencies within this range:
\[
50 + 30 + 10 = 90
\]

So, an estimate for the number of rings that contain between 1.4 g and 3 g of gold is **90 rings**.

### Part (b)
The reason this value is only an estimate is that the histogram provides grouped data, meaning each bar represents a range of values, not exact data points. We assumed all rings within each bar's range are evenly distributed, which may not be precisely accurate. Therefore, the true number could vary slightly from this estimate.

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Would you like more details or have questions on this? Here are some related questions for further practice:

1. How would you estimate the number of rings between 0.5 g and 2.5 g?
2. If the frequency density for the 2.5 g to 3 g range doubled, how would it affect the estimate?
3. How can the width of each bar affect the accuracy of our estimate?
4. What if we needed an estimate for rings above 3 g? How would we approach it?
5. Why might a cumulative frequency graph provide more accurate estimates?

**Tip:** Always consider both the frequency density and width when working with histograms to determine the frequency in each range.

a) Work out an estimate for the number of rings that contain between 1.4 g and 3 g of gold.
b) Write a sentence to explain why the value you calculated in part a) is only an estimate.

Learn how to estimate the number of rings containing between 1.4 g and 3 g of gold using a histogram. This problem covers statistical concepts such as histogram analysis and estimation. Suitable for students in Grades 9-10.

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