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Sagot :
Let's approach this question step by step, starting from solving the formula for [tex]\( r \)[/tex] and then applying it to the given areas.
### Part (a)
The area [tex]\( A \)[/tex] of a circle with radius [tex]\( r \)[/tex] is given by the formula:
[tex]\[ A = \pi r^2 \][/tex]
To solve for [tex]\( r \)[/tex], we need to isolate [tex]\( r \)[/tex] on one side of the equation. Here are the steps to do this:
1. Divide both sides of the equation by [tex]\( \pi \)[/tex] to isolate [tex]\( r^2 \)[/tex]:
[tex]\[ A / \pi = r^2 \][/tex]
2. Take the square root of both sides to solve for [tex]\( r \)[/tex]:
[tex]\[ r = \sqrt{A / \pi} \][/tex]
So our formula for [tex]\( r \)[/tex] in terms of [tex]\( A \)[/tex] and [tex]\( \pi \)[/tex] is:
[tex]\[ r = \sqrt{A / \pi} \][/tex]
### 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 [tex]\( A = 113 \, \text{ft}^2 \)[/tex]:
[tex]\[ r = \sqrt{113 / \pi} \approx 5.997 \][/tex]
Rounding to the nearest unit:
[tex]\[ r \approx 6 \, \text{ft} \][/tex]
2. For [tex]\( A = 1810 \, \text{in}^2 \)[/tex]:
[tex]\[ r = \sqrt{1810 / \pi} \approx 24.003 \][/tex]
Rounding to the nearest unit:
[tex]\[ r \approx 24 \, \text{in} \][/tex]
3. For [tex]\( A = 531 \, \text{m}^2 \)[/tex]:
[tex]\[ r = \sqrt{531 / \pi} \approx 13.001 \][/tex]
Rounding to the nearest unit:
[tex]\[ r \approx 13 \, \text{m} \][/tex]
### Part (c)
Solving the formula for [tex]\( r \)[/tex] in advance is beneficial because it simplifies the calculation process. By isolating [tex]\( r \)[/tex] from the area formula initially, you create a general solution that can be applied to any value of [tex]\( A \)[/tex] 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 [tex]\( A = 113 \, \text{ft}^2 \)[/tex], the radius [tex]\( r \)[/tex] is approximately [tex]\( 6 \, \text{ft} \)[/tex].
- For [tex]\( A = 1810 \, \text{in}^2 \)[/tex], the radius [tex]\( r \)[/tex] is approximately [tex]\( 24 \, \text{in} \)[/tex].
- For [tex]\( A = 531 \, \text{m}^2 \)[/tex], the radius [tex]\( r \)[/tex] is approximately [tex]\( 13 \, \text{m} \)[/tex].
### Part (a)
The area [tex]\( A \)[/tex] of a circle with radius [tex]\( r \)[/tex] is given by the formula:
[tex]\[ A = \pi r^2 \][/tex]
To solve for [tex]\( r \)[/tex], we need to isolate [tex]\( r \)[/tex] on one side of the equation. Here are the steps to do this:
1. Divide both sides of the equation by [tex]\( \pi \)[/tex] to isolate [tex]\( r^2 \)[/tex]:
[tex]\[ A / \pi = r^2 \][/tex]
2. Take the square root of both sides to solve for [tex]\( r \)[/tex]:
[tex]\[ r = \sqrt{A / \pi} \][/tex]
So our formula for [tex]\( r \)[/tex] in terms of [tex]\( A \)[/tex] and [tex]\( \pi \)[/tex] is:
[tex]\[ r = \sqrt{A / \pi} \][/tex]
### 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 [tex]\( A = 113 \, \text{ft}^2 \)[/tex]:
[tex]\[ r = \sqrt{113 / \pi} \approx 5.997 \][/tex]
Rounding to the nearest unit:
[tex]\[ r \approx 6 \, \text{ft} \][/tex]
2. For [tex]\( A = 1810 \, \text{in}^2 \)[/tex]:
[tex]\[ r = \sqrt{1810 / \pi} \approx 24.003 \][/tex]
Rounding to the nearest unit:
[tex]\[ r \approx 24 \, \text{in} \][/tex]
3. For [tex]\( A = 531 \, \text{m}^2 \)[/tex]:
[tex]\[ r = \sqrt{531 / \pi} \approx 13.001 \][/tex]
Rounding to the nearest unit:
[tex]\[ r \approx 13 \, \text{m} \][/tex]
### Part (c)
Solving the formula for [tex]\( r \)[/tex] in advance is beneficial because it simplifies the calculation process. By isolating [tex]\( r \)[/tex] from the area formula initially, you create a general solution that can be applied to any value of [tex]\( A \)[/tex] 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 [tex]\( A = 113 \, \text{ft}^2 \)[/tex], the radius [tex]\( r \)[/tex] is approximately [tex]\( 6 \, \text{ft} \)[/tex].
- For [tex]\( A = 1810 \, \text{in}^2 \)[/tex], the radius [tex]\( r \)[/tex] is approximately [tex]\( 24 \, \text{in} \)[/tex].
- For [tex]\( A = 531 \, \text{m}^2 \)[/tex], the radius [tex]\( r \)[/tex] is approximately [tex]\( 13 \, \text{m} \)[/tex].
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