Answer:
The correct option is;
πR³(2/3 + (1/3)cos³θ - cosθ)
Explanation:
The volume of a segment of a sphere is given by the relation;
[tex]V = \pi \cdot h^2 \cdot \left (R - \dfrac{h}{3} \right)[/tex]
We note that h = R - R·cos(θ)
Therefore by substituting the value of h in the equation of a segment of a sphere, we have;
[tex]V = \pi \cdot \left (R - R\cdot cos(\theta) \right ) ^2 \cdot \left (R - \dfrac{\left (R - R\cdot cos(\theta) \right )}{3} \right)[/tex]
Which gives;
[tex]\dfrac{R^3\cdot \pi \cdot cos^3 (\theta) -3 \cdot R^3 \cdot\pi \cdot cos (\theta) + 2 \cdot R^3 \cdot \pi}{3}[/tex]
[tex]R^3\cdot \pi \cdot \left (\dfrac{cos^3 (\theta) -3 \cdot cos (\theta) + 2 }{3} \right )[/tex]
[tex]R^3\cdot \pi \cdot \left (\dfrac{cos^3 (\theta) + 2 }{3} - cos (\theta) \right )[/tex]
[tex]R^3\cdot \pi \cdot \left (\dfrac{cos^3 (\theta) }{3} + \dfrac{2}{3} - cos (\theta) \right )[/tex]
Therefore, the correct option is πR³(2/3 + (1/3)cos³θ - cosθ).
