To solve for the radius and circumference of a circle given its area, we will use the formulas you provided.
Part 4a: Finding the Radius
The formula for the area \( A \) of a circle is given by:
\[ A = \pi r^2 \]
We are given that the area \( A \) is 1,809.56 square centimeters, so we can set up the equation:
\[ 1,809.56 = \pi r^2 \]
To solve for \( r^2 \), we first divide both sides by \( \pi \):
\[ r^2 = \frac{1,809.56}{\pi} \]
Using the approximate value of \( \pi \) (3.14159), we substitute it into the equation:
\[ r^2 = \frac{1,809.56}{3.14159} \approx 575.00 \]
Next, we find \( r \) by taking the square root of both sides:
\[ r = \sqrt{575.00} \approx 24.08 \text{ cm} \]
Thus, the radius of the circle is approximately 24.08 cm.
Part 4b: Finding the Circumference
The formula for the circumference \( C \) of a circle is:
\[ C = 2\pi r \]
Now that we have found the radius \( r \approx 24.08 \text{ cm} \), we can substitute this value into the circumference formula:
\[ C = 2\pi(24.08) \]
Calculating this gives:
\[ C \approx 2 \times 3.14159 \times 24.08 \approx 150.81 \text{ cm} \]
Thus, the circumference of the circle is approximately 150.81 cm.
Summary of Results:
- Radius: \( \approx 24.08 \text{ cm} \)
- Circumference: \( \approx 150.81 \text{ cm} \)
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