Question

From a solid cylinder of height 30 cm and radius 7 cm, a conical cavity of height 24 cm and of base radius 7 cm is drilled out. Find the volume and the total surface of the remaining solid. [4 MARKS]

Answer 1

Radius, r = 7 cm.

Height of the cylinder, H = 30 cm.

Height of the cone, h = 24 cm.

Slant height of the cone,
$l=\sqrt{h^2+r^2}=\sqrt{(24{)}^2+(7{)}^2}=\sqrt{625}=25cm$

(i) Volume of the remaining solid

= (Volume of the cylinder) - (Volume of the cone)

$=\pi r^{2}H-\frac{1}{3}\pi r^{2}h=\pi r^2(H-\frac{h}{3})$

$=[\frac{22}{7}\times 7\times 7\times (30-\frac{24}{3})]c{m}^3$

$=[\frac{22}{7}\times 7\times 7\times 22]c{m}^3$

$=(22\times 7\times 22)c{m}^3=3388c{m}^3$.

(ii) Total surface area of the remaining solid

= Curved surface area of cylinder + Curved surface area of cone + Area of (upper) circular base of cylinder

$=2\pi rH+\pi rl+\pi r^2=\pi r(2H+l+r)c{m}^2$

$=\frac{22}{7}\times 7\times (60+25+7)$

$=(22\times 92)c{m}^2$

$=2024c{m}^2$.