To find the radius of a sphere given its surface area, we will 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] (pi) is a constant approximately equal to 3.14159
- [tex]\( r \)[/tex] is the radius of the sphere
Given the surface area [tex]\( A = 452.4 \, \text{cm}^2 \)[/tex], we need to solve for [tex]\( r \)[/tex].
Step-by-step process:
1. Substitute the given surface area into the surface area formula:
[tex]\[ 452.4 = 4 \pi r^2 \][/tex]
2. Isolate [tex]\( r^2 \)[/tex] by dividing both sides of the equation by [tex]\( 4\pi \)[/tex]:
[tex]\[ r^2 = \frac{452.4}{4 \pi} \][/tex]
3. Calculate the quantity on the right-hand side:
[tex]\[ r^2 = \frac{452.4}{4 \times 3.14159} \][/tex]
[tex]\[ r^2 \approx 36.000848127386725 \][/tex]
4. Take the square root of both sides to find [tex]\( r \)[/tex]:
[tex]\[ r = \sqrt{36.000848127386725} \][/tex]
[tex]\[ r \approx 6.000070676865959 \][/tex]
Thus, the radius [tex]\( r \)[/tex] of the sphere is approximately [tex]\( 6.000 \, \text{cm} \)[/tex].
Therefore, the radius of the sphere that has a surface area of [tex]\( 452.4 \, \text{cm}^2 \)[/tex] is approximately [tex]\( 6.000 \, \text{cm} \)[/tex].
