To solve both parts of the problem, we will use the relationships between circumference, diameter, radius, and area of a circle.
5a. Calculate the Diameter
The formula for the circumference \( C \) of a circle is given by:
\[ C = \pi d \]
Where:
- \( C \) is the circumference,
- \( d \) is the diameter,
- \( \pi \) (pi) is approximately \( 3.14159 \).
We know the circumference \( C \) is 131.95 meters, so we can rearrange the formula to solve for the diameter \( d \):
\[ d = \frac{C}{\pi} \]
Plugging in the value of \( C \):
\[ d = \frac{131.95}{\pi} \approx \frac{131.95}{3.14159} \approx 42.00 \text{ meters} \]
Thus, the diameter of the circle is approximately 42.00 meters.
5b. Calculate the Area
To calculate the area \( A \) of the same circle, we use the formula:
\[ A = \pi r^2 \]
Where:
- \( A \) is the area,
- \( r \) is the radius of the circle.
Since the radius \( r \) is half of the diameter \( d \):
\[ r = \frac{d}{2} = \frac{42.00}{2} = 21.00 \text{ meters} \]
Now we can plug the radius into the area formula:
\[ A = \pi r^2 = \pi (21.00)^2 \]
Calculating \( (21.00)^2 \):
\[ (21.00)^2 = 441.00 \]
Now substituting back into the area formula:
\[ A = \pi \times 441.00 \approx 3.14159 \times 441.00 \approx 1385.44 \text{ square meters} \]
Thus, the area of the circle is approximately 1385.44 square meters.
Summary:
- Diameter: 42.00 meters
- Area: 1385.44 square meters
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