Question 14 Marks
If the ratio of radius of base and height of a cone is 5 : 12 and its volume is 314 cubic metre. Find its perpendicular height and slant height (π = 3.14).
Given: Ratio of radius of base and height of a cone = 5 : 12,
Volume = 314 cubic metre
To find: Perpendicular height (h) and slant height (l)
Given: Ratio of radius of base and height of a cone = 5 : 12,
Volume = 314 cubic metre
To find: Perpendicular height (h) and slant height (l)
Answer
View full question & answer→i. The ratio of radius and height of cone is $5 : 12$
Let the common multiple be $x.$
∴ Radius of base $(r) = 5x$
Perpendicular height $(h) = 12x$
$\text { Volume of cone }=\frac{1}{3} \pi r^2 h$
$\therefore \quad 314=\frac{1}{3} \times 3.14 \times(5 x)^2 \times(12 x)$
$\therefore \quad 314=\frac{1}{3} \times 3.14 \times 25 x^2 \times 12 x$
$\therefore \quad x^3=\frac{314 \times 3}{3.14 \times 25 \times 12}$
$=\frac{314 \times 3 \times 100}{314 \times 25 \times 12}$
$\therefore x^3 = 1$
$\therefore x = 1 $… [Taking cube root on both sides]
\therefore r = 5x = 5(1) = 5m
$h = 12x = 12(1) = 12 mii$. Now,$l^2 = r^2 + h^2$
$= 5^2 + 12^2$
$= 25 + 144$
$\therefore l^2 = 169$
$\therefore I =\sqrt{169} \ldots$ [Taking square root on both sides]
$= 13 m$
The perpendicular height and slant height of the cone are $12 m$ and $13 m$ respectively.
Let the common multiple be $x.$
∴ Radius of base $(r) = 5x$
Perpendicular height $(h) = 12x$
$\text { Volume of cone }=\frac{1}{3} \pi r^2 h$
$\therefore \quad 314=\frac{1}{3} \times 3.14 \times(5 x)^2 \times(12 x)$
$\therefore \quad 314=\frac{1}{3} \times 3.14 \times 25 x^2 \times 12 x$
$\therefore \quad x^3=\frac{314 \times 3}{3.14 \times 25 \times 12}$
$=\frac{314 \times 3 \times 100}{314 \times 25 \times 12}$
$\therefore x^3 = 1$
$\therefore x = 1 $… [Taking cube root on both sides]
\therefore r = 5x = 5(1) = 5m
$h = 12x = 12(1) = 12 mii$. Now,$l^2 = r^2 + h^2$
$= 5^2 + 12^2$
$= 25 + 144$
$\therefore l^2 = 169$
$\therefore I =\sqrt{169} \ldots$ [Taking square root on both sides]
$= 13 m$
The perpendicular height and slant height of the cone are $12 m$ and $13 m$ respectively.