Question

A toy is in the form of a hemisphere surmounted by a right circular cone of the same base radius as that of the hemisphere.If the radius of the base of the cone is 21 cm and its volume is $\frac{2}{3}$ of the volume of the hemisphere,calculate the height of the cone and the surface area of the toy? $(Use\pi =22/7)$

Answer 1

Solution:

As per the parameter given in the question itself, we have

Radius of the cone = 21 cm = R

Radius of the hemisphere = Radius of the cone = 21 cm

Volume of the cone = $\frac{2}{3}$ of the hemisphere

We know that,

The volume of the cone = V₁ = $\frac{1}{3}\pi R^{2}L=\frac{1}{3}\times \pi \times {21}^{2}L$

Also, we know that,

The volume of the hemisphere = V₂ = $\frac{2}{3}\pi R^3=\frac{2}{3}\times \pi \times {21}^3$cm³

Now, as per the condition

V₁ = $\frac{2}{3}{V}_2$

V₁ = 23$\times$169714.286

$\frac{1}{3}\times \pi \times {21}^{2}L=\frac{2}{3}\times \pi \times {21}^3$

L = 28 cm

Curved surface area of the Cone = CSA₁ = $\pi RL=\pi \times 21\times 28$cm²

Curved Surface area of the hemisphere = CSA₂ = $2\pi R^2=2\pi \times {21}^2$ cm²

Now, the total surface area = S = CSA₁ + CSA₂

S = $\pi \times 21\times 28+2\pi \times {21}^2$

S = 5082 cm²

Therefore, the curved surface area of the toy = S = 5082 cm²