Answer :
Sure, let's tackle the problem step-by-step:
### Part (a): Calculating the Radius of the Sphere
To find the radius of a sphere given its surface area, we can use the formula for the surface area of a sphere:
[tex]\[ A = 4 \pi r^2 \][/tex]
where [tex]\( A \)[/tex] is the surface area, [tex]\( \pi \)[/tex] is approximately 3.14, and [tex]\( r \)[/tex] is the radius.
Given:
[tex]\[ A = 9244 \text{ square feet} \][/tex]
[tex]\[ \pi = 3.14 \][/tex]
1. Rearrange the formula to solve for [tex]\( r \)[/tex]:
[tex]\[ 4 \pi r^2 = A \][/tex]
2. Divide both sides by [tex]\( 4 \pi \)[/tex] to isolate [tex]\( r^2 \)[/tex]:
[tex]\[ r^2 = \frac{A}{4 \pi} \][/tex]
3. Substitute the given values:
[tex]\[ r^2 = \frac{9244}{4 \times 3.14} \][/tex]
4. Calculate the value inside the fraction:
[tex]\[ r^2 = \frac{9244}{12.56} \][/tex]
[tex]\[ r^2 \approx 736 \][/tex]
5. Take the square root of both sides to find [tex]\( r \)[/tex]:
[tex]\[ r = \sqrt{736} \][/tex]
[tex]\[ r \approx 27.13 \][/tex]
So, the radius of the sphere is approximately 27.13 feet.
### Part (b): Checking Your Answer
To verify our calculated radius, we can recompute the surface area using this radius and see if it matches the given surface area.
1. Use the surface area formula:
[tex]\[ A = 4 \pi r^2 \][/tex]
2. Substitute [tex]\( r \approx 27.13 \)[/tex] and [tex]\( \pi = 3.14 \)[/tex]:
[tex]\[ A = 4 \times 3.14 \times (27.13)^2 \][/tex]
3. Calculate [tex]\( (27.13)^2 \)[/tex]:
[tex]\[ (27.13)^2 \approx 736 \][/tex]
4. Substitute and compute the surface area:
[tex]\[ A = 4 \times 3.14 \times 736 \][/tex]
[tex]\[ A = 12.56 \times 736 \][/tex]
[tex]\[ A \approx 9244 \][/tex]
The calculated surface area is approximately 9244 square feet, which matches the given surface area. Thus, our radius calculation is confirmed to be correct.
The radius of the sphere is approximately 27.13 feet and the surface area check verifies this result.
### Part (a): Calculating the Radius of the Sphere
To find the radius of a sphere given its surface area, we can use the formula for the surface area of a sphere:
[tex]\[ A = 4 \pi r^2 \][/tex]
where [tex]\( A \)[/tex] is the surface area, [tex]\( \pi \)[/tex] is approximately 3.14, and [tex]\( r \)[/tex] is the radius.
Given:
[tex]\[ A = 9244 \text{ square feet} \][/tex]
[tex]\[ \pi = 3.14 \][/tex]
1. Rearrange the formula to solve for [tex]\( r \)[/tex]:
[tex]\[ 4 \pi r^2 = A \][/tex]
2. Divide both sides by [tex]\( 4 \pi \)[/tex] to isolate [tex]\( r^2 \)[/tex]:
[tex]\[ r^2 = \frac{A}{4 \pi} \][/tex]
3. Substitute the given values:
[tex]\[ r^2 = \frac{9244}{4 \times 3.14} \][/tex]
4. Calculate the value inside the fraction:
[tex]\[ r^2 = \frac{9244}{12.56} \][/tex]
[tex]\[ r^2 \approx 736 \][/tex]
5. Take the square root of both sides to find [tex]\( r \)[/tex]:
[tex]\[ r = \sqrt{736} \][/tex]
[tex]\[ r \approx 27.13 \][/tex]
So, the radius of the sphere is approximately 27.13 feet.
### Part (b): Checking Your Answer
To verify our calculated radius, we can recompute the surface area using this radius and see if it matches the given surface area.
1. Use the surface area formula:
[tex]\[ A = 4 \pi r^2 \][/tex]
2. Substitute [tex]\( r \approx 27.13 \)[/tex] and [tex]\( \pi = 3.14 \)[/tex]:
[tex]\[ A = 4 \times 3.14 \times (27.13)^2 \][/tex]
3. Calculate [tex]\( (27.13)^2 \)[/tex]:
[tex]\[ (27.13)^2 \approx 736 \][/tex]
4. Substitute and compute the surface area:
[tex]\[ A = 4 \times 3.14 \times 736 \][/tex]
[tex]\[ A = 12.56 \times 736 \][/tex]
[tex]\[ A \approx 9244 \][/tex]
The calculated surface area is approximately 9244 square feet, which matches the given surface area. Thus, our radius calculation is confirmed to be correct.
The radius of the sphere is approximately 27.13 feet and the surface area check verifies this result.
