https://file.aiquickdraw.com/mathbot/file/1726364862594bmutbwnx.jpg

Let's break down the problem shown in the image step by step.

### Part (a): Finding the ratio of Circumference to Diameter

The relationship between the circumference \( C \) and diameter \( d \) of any circle is given by the formula:

\[
\frac{C}{d} = \pi
\]

We can calculate the ratio \( \frac{C}{d} \) for each circle using the provided measurements:

#### Circle 1:
- Circumference \( C_1 = 47.121 \, \text{units} \)
- Diameter \( d_1 = 15 \, \text{units} \)

\[
\frac{C_1}{d_1} = \frac{47.121}{15} = 3.1414
\]

#### Circle 2:
- Circumference \( C_2 = 25.128 \, \text{units} \)
- Diameter \( d_2 = 8 \, \text{units} \)

\[
\frac{C_2}{d_2} = \frac{25.128}{8} = 3.141
\]

#### Circle 3:
- Circumference \( C_3 = 39.254 \, \text{units} \)
- Diameter \( d_3 = 12.5 \, \text{units} \)

\[
\frac{C_3}{d_3} = \frac{39.254}{12.5} = 3.1403
\]

### Part (b): Choosing the correct formula

The formula that relates the circumference \( C \) and the diameter \( d \) is:

\[
C = \pi d
\]

From the given options, the correct formula is:

\[
\frac{C}{d} = \pi
\]

So the correct answer is the second option: \( \frac{C}{d} = \pi \).

### Part (c): Formula for Circumference in terms of Diameter

The formula that relates the circumference \( C \) of a circle to its diameter \( d \) is:

\[
C = \pi d
\]

This is the standard formula for the circumference of a circle.

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Do you want more details or have any questions? Here are 5 related questions for further exploration:

1. How would the circumference change if the diameter of Circle 1 were doubled?
2. What would the ratio \( \frac{C}{d} \) be for a circle with a diameter of 20 units?
3. How does the circumference of a circle relate to its radius?
4. Can you derive the formula \( C = \pi d \) from the definition of \( \pi \)?
5. What would be the area of Circle 2?

### Tip: Remember, the constant \( \pi \) (approximately 3.1416) is the ratio of the circumference of any circle to its diameter!

Below are three circles. The circumference and diameter of each circle have been approximately measured. Use the measurements and the calculator to fill in the blanks, choose the correct formula, and write a formula for circumference.

This problem involves calculating the ratio of circumference to diameter for three circles and determining the correct formula involving pi. The solution shows how to calculate the ratio for different circles and explains the standard formula C = π * d.

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