Radius is the only property of a sphere that is used to define surface areas and volumes. We need to find what the radius is from the information that the question has given us.
S is a quarter sphere, and we have been given its volume. This means that we can work out the volume of a whole sphere with the same radius as S by doing the calculation
4 x 576(pi)cm³ = 2304(pi)cm³
We know that the volume of a sphere is given by (4/3)(pi)(r³), so we can say that
2304(pi)cm³ = (4/3)(pi)(r³)
Where r is the radius of S
Rearranging gives us
r³ = 1728cm³ r = 12cm
Now we can calculate the surface area of S
S has two semi-circular faces, and one quarter-spherical face.
The area of a circle is equal to (pi)r², and two semi-circles make one circle so the surface area of the two semi-circular faces is:
(pi)((12cm)²) 144(pi)cm²
The surface area of a sphere is given by 4(pi)r², and therefore the surface area of a quarter-sphere is also (pi)r², which gives an answer of:
144(pi)cm²
The total surface area of S is the sum of the surface areas of all faces of S:
144(pi)cm² + 144(pi)cm² = 288(pi)cm²
And now we can substitute the value of pi in to find that the surface area of S is:
904.78cm²
Rounding to 3 significant figures the answer is:
905cm²

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