Let's approach this question step by step, starting from solving the formula for \(\( r \)\) and then applying it to the given areas.
### Part (a)
The area \(\( A \)\) of a circle with radius \(\( r \)\) is given by the formula:
\(\[ A = \pi r^2 \]\)
To solve for \(\( r \)\), we need to isolate \(\( r \)\) on one side of the equation. Here are the steps to do this:
1. Divide both sides of the equation by \(\( \pi \)\) to isolate \(\( r^2 \)\):
\(\[ A / \pi = r^2 \]\)
2. Take the square root of both sides to solve for \(\( r \)\):
\(\[ r = \sqrt{A / \pi} \]\)
So our formula for \(\( r \)\) in terms of \(\( A \)\) and \(\( \pi \)\) is:
\(\[ r = \sqrt{A / \pi} \]\)
### Part (b)
Now, we will use this formula to find the radius for each given area, rounding each result to the nearest unit.
1. For \(\( A = 113 \, \text{ft}^2 \)\):
\(\[
r = \sqrt{113 / \pi} \approx 5.997
\]\)
Rounding to the nearest unit:
\(\[
r \approx 6 \, \text{ft}
\]\)
2. For \(\( A = 1810 \, \text{in}^2 \)\):
\(\[
r = \sqrt{1810 / \pi} \approx 24.003
\]\)
Rounding to the nearest unit:
\(\[
r \approx 24 \, \text{in}
\]\)
3. For \(\( A = 531 \, \text{m}^2 \)\):
\(\[
r = \sqrt{531 / \pi} \approx 13.001
\]\)
Rounding to the nearest unit:
\(\[
r \approx 13 \, \text{m}
\]\)
### Part (c)
Solving the formula for \(\( r \)\) in advance is beneficial because it simplifies the calculation process. By isolating \(\( r \)\) from the area formula initially, you create a general solution that can be applied to any value of \(\( A \)\) effortlessly. This helps in quickly finding the radius for different areas without re-deriving the formula each time. It saves time and reduces the probability of calculation errors, providing a clear and straightforward method to find the radius from the given area.
So, the summarized solutions are:
- For \(\( A = 113 \, \text{ft}^2 \)\), the radius \(\( r \)\) is approximately \(\( 6 \, \text{ft} \)\).
- For \(\( A = 1810 \, \text{in}^2 \)\), the radius \(\( r \)\) is approximately \(\( 24 \, \text{in} \)\).
- For \(\( A = 531 \, \text{m}^2 \)\), the radius \(\( r \)\) is approximately \(\( 13 \, \text{m} \)\).
