The volume of a right circular cone is 9856 cm³. If the diameter of the base is 28 cm, find

(i) height of the cone

(ii) slant height of the cone

(iii) curved surface area of the cone

Given,

Volume of cone = 9856 cm³

Diameter of the cone (d) = 28 cm

Radius of the cone (r) = $\dfrac{\text{Diameter}}{2} = \dfrac{28}{2}$ = 14 cm.

(i) Let height of cone be h cm.

Volume of the cone = 9856 cm³

Substituting values we get :

$\Rightarrow \dfrac{1}{3}πr^2h = 9856 \\[1em] \Rightarrow h = 9856 \times \dfrac{3}{πr^2} \\[1em] \Rightarrow h = 9856 \times \dfrac{3}{\dfrac{22}{7} \times 14^2} \\[1em] \Rightarrow h = 9856 \times \dfrac{3 \times 7}{22 \times 196} \\[1em] \Rightarrow h = \dfrac{206976}{4312} \\[1em] \Rightarrow h = 48 \text{ cm}.$

Hence, the height of the cone = 48 cm.

(ii) Let slant height of the cone be l cm.

By formula,

$\Rightarrow l = \sqrt{r^2 + h^2} \\[1em] \Rightarrow l = \sqrt{(14)^2 + (48)^2} \\[1em] \Rightarrow l = \sqrt{196 + 2304} \\[1em] \Rightarrow l = \sqrt{2500} = 50 \text{ cm}.$

Hence, the slant height of the cone = 50 cm.

(iii) By formula,

Curved surface area of cone = πrl

Substituting values we get :

⇒ Curved surface area of cone = $\dfrac{22}{7}$ × 14 × 50

= 22 x 2 x 50

= 44 x 50

= 2200 cm².

Hence, curved surface area of cone = 2200 cm².