Question

A bucket is in the form of a frustum of a cone with a capacity of 12308.8 $c{m}^3$ of water. The radii of the top and bottom circular ends are 20 cm and 12 cm respectively. Find the height of the bucket and the area of the metal sheet used in its making. (Use $\pi$ = 3.14)

Answer 1

Soln:

Radii of top circular ends $r_1=20cm$

Radii of bottom circular end of bucket $r_2=12cm$

Let height of bucket be ‘h’

Volume of frustum cone $=\frac{1}{3}\pi (r_1^2+r_2^2+r_1{r}_2)h$
$=\frac{1}{3}\pi ({20}^2+{12}^2+20\times 12)h=\frac{784\pi h}{3}c{m}^3$ —-(a)

Given capacity/ volume of the bucket = 12308.8 $c{m}^3$—–(b)

Equating (a) and (b)

$\frac{784\pi h}{3}=1208.8$

$h=\frac{1208.8\times 3}{784\pi }=15cm$

∴∴ height of the bucket (h) = 15 cm

Let ‘L’ be slant height of bucket

$L^2=(r_1-r_2{)}^2+h^2$

$L=\sqrt{(r_1-r_2{)}^2+h^2}=17cm$

∴ length of the bucket/ slant height of the bucket (L) =17 cm

Curved surface area of bucket =

$\pi (r_1+r_2)L+\pi r_2^2$

$\pi (20+12)17+\pi \times {12}^2=\pi (9248+144)=2160.32c{m}^3$

∴ curved surface area = 2160.32 cm²