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.

• A. Volume $=2024c{m}^3$
• B. Surface Area $=3388c{m}^2$
• C. Volume $=3388c{m}^3$
• D. Surface Area $=2024c{m}^2$

Answer 1

Answer: (C), (D)

Volume $=3388c{m}^3$
Surface Area $=2024c{m}^2$

Radius, $r=7cm$

Height of the cylinder, $H=30cm$

Height of the cone, $h=24cm$

Slant height of the cone

$\iota =\sqrt{h^2+r^2}=\sqrt{(24{)}^2+(7{)}^2}=\sqrt{625}=25cm$

(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

= $2\pi rH+\pi rl+\pi r^2=\pi r(2H+l+r)$

= $[\frac{22}{7}\times 7times(60+25+7\left)\right]c{m}^2$

= $(22\times 92)c{m}^2=2024c{m}^2$