Question

A girl empties a cylindrical bucket, full of sand, of base radius 18 cm and height 32 cm, on the floor to form a conical heap of sand. If the height of this conical heap is 24 cm, then find its slant height correct upto one place of decimal?

Answer 1

Given:
Radius (r) of the cylindrical bucket = 18 cm
Height (h) of the cylindrical bucket = 32 cm
Volume of the cylindrical bucket = \[\pi r^2 h\]

\[= \left( \pi \times 18 \times 18 \times 32 \right) {cm}^3\]

Let R be the radius of the conical heap.
Height (H) of the conical heap = 24 cm
Volume of the conical heap =\[\frac{1}{3}\pi R^2 H = \left( \frac{1}{3}\pi R^2 \times 24 \right) {cm}^3 = 8\pi R^2 {cm}^3\]

According to the question:
Volume of the cylindrical bucket = Volume of the conical heap

\[\Rightarrow \pi \times 18 \times 18 \times 32 = 8 \times \pi \times R^2 \]

\[ \Rightarrow R^2 = \frac{18 \times 18 \times 32}{8}\]

\[ = 18\]\[\times\]\[18\]\[\times\]\[4\]\[\]\[\]\[\]\[\]\[\]\[\]\[\]\[\]\[\]\[\]\[\]\[=\]\[1296\]

\[\Rightarrow R = 36\]

Let l be the slant height of the conical heap.

\[\therefore l^2 = R^2 + H^2 \]

\[ = {36}^2 + {24}^2 \]

\[ = 1296 + 576\]

\[ = 1872\]

\[ \Rightarrow l = 43 . 3\]

Thus, the slant height (l) of the conical heap is 43.3 cm.