Question

Which of the following is the volume of the shaded in Of the sphere?

Answer 1

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;

$V = \pi \cdot h^2 \cdot \left (R - \dfrac{h}{3} \right)$

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;

$V = \pi \cdot \left (R - R\cdot cos(\theta) \right ) ^2 \cdot \left (R - \dfrac{\left (R - R\cdot cos(\theta) \right )}{3} \right)$

Which gives;

$\dfrac{R^3\cdot \pi \cdot cos^3 (\theta) -3 \cdot R^3 \cdot\pi \cdot cos (\theta) + 2 \cdot R^3 \cdot \pi}{3}$

$R^3\cdot \pi \cdot \left (\dfrac{cos^3 (\theta) -3 \cdot cos (\theta) + 2 }{3} \right )$

$R^3\cdot \pi \cdot \left (\dfrac{cos^3 (\theta) + 2 }{3} - cos (\theta) \right )$

$R^3\cdot \pi \cdot \left (\dfrac{cos^3 (\theta) }{3} + \dfrac{2}{3} - cos (\theta) \right )$

Therefore, the correct option is πR³(2/3 + (1/3)cos³θ - cosθ).