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Examples 36 to 40 of Mensuration

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(36) What is the volume of the right circular cone in the given figure ?

Soln.

Volume = $\frac{1}{3}$ r²h

= $\frac{1}{3} \times \frac{22}{7} \times 6 \times 6 \times 7$

= 264 cm³.

(37) Find the slant height, volume, curved surface area and the whole surface area of a cone of base radius 21 m and height 28 m.

Soln. Here, r = 21 cm and h = 28 cm.

Slant height, l = $\sqrt{r^2+h^2} = \sqrt{\left(21\right)^2+\left(28\right)^2}$ = 35 m.

Volume = $\frac{1}{3}\pi$ r²h = $\frac{1}{3}\times\frac{22}{7}\times21\times21\times28$ = 12936 cm³.

Curved surface area = πr l = $\frac{22}{7}\times21\times35$ = 2310 cm².

Total surface area = (πr l + πr² ) = $\left (2310 + \frac{22}{7} \times 21 \times 21 \right )$ = 3696 cm².

(38) A piece of metal in the form of a cone of radius 44 cm and height 42 cm is converted to a cube. Find the side of the cube.

Soln. Volume of a Cube

∴ x³ = Volume of a cone

= $\frac{1}{3}\pi$ r²h

= $\frac{1}{3} \times \frac{22}{7} \times 44 \times 44 \times 42$

= $44 \times 44 \times 44$

∴ x = Side of the cube = 44 cm.

(39) Find the area of the iron sheet required to prepare a cone 13 m high with base radius 12 m.

Soln. Here, r = 7 m & h = 24 m.

Now, l = $\sqrt{\left(24\right)^2+\left(7\right)^2} = \sqrt{625}$ = 25 m.

∴ Area of the sheet = Total surface area of the cone.

= π r ( l + r )

= $\frac{22}{7} \times 7 \times ( 25 + 7 )$

= 704 m².

(40) Find the volume and surface area of a sphere of diameter 42 cm.

Soln. Volume = $\frac{4}{3}$ r³ = $\frac{4}{3} \times \times 21 \times 21 \times 21$ = 38808 cm³.

Curved surface area = 4πr² = $4 \times \frac{22}{7} \times 21 \times 21$ = 5544 cm².