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Find the total surface area of a cone, if its slant height is 21 m and diameter of its base is 24 m.
Answer: Slant height ($l$) = 21 m. Radius ($r$) = $24/2 = 12$ m.
Total Surface Area (TSA) = $\pi r(l + r)$
TSA = $\frac{22}{7} \times 12 \times (21 + 12) = \frac{22}{7} \times 12 \times 33 = \frac{8712}{7} \approx 1244.57$ m².
The height of a cone is 16 cm, and its base radius is 12 cm. Find the curved surface area and the total surface area of the cone.
Answer: Height ($h$) = 16 cm, Radius ($r$) = 12 cm.
First, find slant height ($l$): $l = \sqrt{h^2 + r^2} = \sqrt{16^2 + 12^2} = \sqrt{256 + 144} = \sqrt{400} = 20$ cm.
Curved Surface Area (CSA) = $\pi rl = 3.14 \times 12 \times 20 = 753.6$ cm² (or exactly $240\pi$ cm²).
Total Surface Area (TSA) = $\pi r(l + r) = 3.14 \times 12 \times (20 + 12) = 3.14 \times 12 \times 32 = 1205.76$ cm² (or exactly $384\pi$ cm²).
A cone has a height of 15 cm. If its volume is 1570 cm³, find the radius of the base.
Answer: Volume ($V$) = $\frac{1}{3}\pi r^2 h$.
$1570 = \frac{1}{3} \times 3.14 \times r^2 \times 15$
$1570 = 5 \times 3.14 \times r^2$
$1570 = 15.7 \times r^2$
$r^2 = 100 \Rightarrow r = 10$ cm.
A right triangle with sides 6 cm, 8 cm and 10 cm is rotated through 360° about the side of 8 cm. Find the volume and the curved surface area of the solid so formed.
Answer: Rotating the triangle about the 8 cm side forms a right circular cone with height ($h$) = 8 cm, base radius ($r$) = 6 cm, and slant height ($l$) = 10 cm.
Volume ($V$) = $\frac{1}{3}\pi r^2 h = \frac{1}{3} \times 3.14 \times 36 \times 8 = 301.44$ cm³ (or exactly $96\pi$ cm³).
Curved Surface Area (CSA) = $\pi rl = 3.14 \times 6 \times 10 = 188.4$ cm² (or exactly $60\pi$ cm²).
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