Question

A metallic toy in the form of a cone of radius 11 cm and height 62 cm mounted on a hemisphere of the same radius is melted and recast into a solid cube. Find the surface area of the cube this formed.

Answer 1

Step I Given Terms

Radius of Cone ($r_1$) = 11 Cm

Height of Cone (h) = 62 Cm

Radius of Hemisphere ($r_2$) = 11 Cm

Step II Finding the Volume of Cone

We know that, Volume of Cone = $\frac{1}{3}{{\mathrm{\pi r}}_1}^2\mathrm{h}$

Substituting the values, we get

Volume of Cone = $\frac{1}{3}\mathrm{\pi }{\left(11\right)}^2\times 62$

= $\frac{1}{3}\mathrm{\pi }\times 11\times 11\times 62C{m}^3$

Step III Finding the Volume of Hemisphere

We know that, Volume of Hemisphere = $\frac{2}{3}{{\mathrm{\pi r}}_2}^3$

Substituting the values we get

Volume of Hemisphere = $\frac{2}{3}\times {\pi \times \left(11Cm\right)}^3$

⇒ Volume of Hemisphere = $\frac{2}{3}\times \mathrm{\pi }\times 11\times 11\times 11{\mathrm{Cm}}^3$

Step IV Finding the Total Volume of the Toy

Total Volume of the Toy = Volume of Hemisphere + Volume of Cone

Substituting the values, we get

Total Volume of the Toy = $\frac{2}{3}\times \mathrm{\pi }\times 11\times 11\times 11{\mathrm{Cm}}^3+$$\frac{1}{3}\mathrm{\pi }\times 11\times 11\times 62C{m}^3$

⇒ Total Volume of Toy = $\frac{1}{3}\mathrm{\pi }\times 11\times 11\left(22+62\right)C{m}^3$

⇒ Total Volume of Toy = $\frac{1}{3}\mathrm{\pi }\times 121\times 84C{m}^3$

Step V Finding the Volume of Cube

We know that, Volume of Cube = $a^3$ (where a = edge length)

Since Cube is made from the toy, therefore Volume of Cube = Total Volume of Toy

Substituting the values we get

$a^3=$$\frac{1}{3}\mathrm{\pi }\times 121\times 84C{m}^3$

⇒ $a^3=\frac{1}{3}\times \frac{22}{7}\times 121\times 84C{m}^3$ {Using $\mathrm{\pi }=\frac{22}{7}$}

Upon Simplification, we get

$a^3=22\times 121\times 4C{m}^3$

⇒ $a^3=10648C{m}^3$

⇒ $a^{}=\sqrt[3]{10648}Cm$

⇒ $a=22Cm$

Step VI Finding the Surface Area of Cube

We know that, Surface area of Cube = $6a^2$

Substituting the values, we get

Surface area of Cube = $6\times {\left(22Cm\right)}^2$

⇒ Surface area of Cube = $2904C{m}^2$

Hence, Surface area of Cube is $2904C{m}^2$