Westonci.ca is the premier destination for reliable answers to your questions, provided by a community of experts. Get detailed and precise answers to your questions from a dedicated community of experts on our Q&A platform. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
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].
We hope our answers were useful. Return anytime for more information and answers to any other questions you have. We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Get the answers you need at Westonci.ca. Stay informed with our latest expert advice.