Sure, let's solve this problem step-by-step.
### 4.1.1 Calculate the radius of the sphere.
Given:
[tex]\[ \text{Surface Area of the sphere} = 6400\pi \, \text{cm}^2 \][/tex]
We know the formula for the surface area of a sphere is:
[tex]\[ 4\pi r^2 \][/tex]
Here, [tex]\( r \)[/tex] is the radius of the sphere. To find the radius, we'll set up the equation:
[tex]\[ 4\pi r^2 = 6400\pi \][/tex]
Now, divide both sides of the equation by [tex]\( 4\pi \)[/tex]:
[tex]\[ r^2 = \frac{6400\pi}{4\pi} \][/tex]
Simplify the right-hand side:
[tex]\[ r^2 = \frac{6400}{4} \][/tex]
[tex]\[ r^2 = 1600 \][/tex]
To find [tex]\( r \)[/tex], take the square root of both sides:
[tex]\[ r = \sqrt{1600} \][/tex]
[tex]\[ r = 40 \, \text{cm} \][/tex]
So, the radius of the sphere is [tex]\( 40 \, \text{cm} \)[/tex].
### 4.1.2 Calculate the volume of the sphere.
Given:
[tex]\[ r = 40 \, \text{cm} \][/tex]
We know the formula for the volume of a sphere is:
[tex]\[ V = \frac{4}{3} \pi r^3 \][/tex]
Substitute [tex]\( r = 40 \)[/tex]:
[tex]\[ V = \frac{4}{3} \pi (40)^3 \][/tex]
Calculate [tex]\( (40)^3 \)[/tex]:
[tex]\[ (40)^3 = 64000 \][/tex]
Now substitute back into the volume formula:
[tex]\[ V = \frac{4}{3} \pi \times 64000 \][/tex]
Simplify:
[tex]\[ V = \frac{4 \times 64000}{3} \pi \][/tex]
[tex]\[ V = \frac{256000}{3} \pi \][/tex]
Now, substituting [tex]\( \pi \)[/tex] back in, we get:
[tex]\[ V \approx 268082.573106329 \, \text{cm}^3 \][/tex]
So, the volume of the sphere is approximately [tex]\( 268082.573 \, \text{cm}^3 \)[/tex].
