Question

A vessel is in the form of an inverted cone. Its height is 8 cm and the radius of its top, which is open, is 5 cm. It is filled with water up to the brim. When lead shots, each of which is a sphere of radius 0.5 cm are dropped into the vessel, one-fourth of the water flows out. Find the number of leads shots dropped in the vessel.

Answer 1

$\huge\boxed{\bf{\red{Solution:-}}}$

$\bf{For \: cone}$

$\sf \: Radius \: of \: the \: top \: (r) = 5 \: cm$

$\sf \: Height \: (h) = 8 \: cm$

$\therefore \: \: \: \: \: \sf \: \: Volume \: of \: the \: cone$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \sf = \frac{1}{3}\pi r {}^{2}h = \frac{1}{3}\pi(5 ) {}^{2} 8$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \sf = \frac{1}{3}\pi \times 25 \times 8 = \frac{200}{3}\pi \: cm {}^{3}$

$\therefore \: \: \: \: \: \sf \: Volume \: of \: water \: in \: cone$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: = \sf \frac{200\pi}{3} \: cm {}^{3}$

$\sf \: [∵ \: the \:cone \:is \: filled \: \:with \: water\: upto\: the\:brim \:(i.e., \: top)]$

$\: \: \: \bf{For \: spherical \: lead \: shot}$

$\sf \: Radius \: (R) = 0.5 \: cm = \frac{5}{10} = \frac{1}{2} \: cm$

$\therefore \: \: \: \: \: \: \: \sf \: Volume \: of \: a \: spherical \: lead \: shot$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \sf = \: \frac{4}{3}\pi R {}^{3} = \frac{4}{3}\pi \bigg( \frac{1}{2} \bigg) {}^{2} = \frac{4\pi}{3} \times \frac{1}{8}$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \sf = \: \frac{\pi}6 \: cm {}^{3} \: \: \: \: \: \: \: \: \: \: ...(1)$

$\bf{Given :} \: \sf \: Volume \: of \: water \: that \: flows \: out \:$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: = \sf \frac{1}{4 } \: volume \: of \: the \: (water) \: cone$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \sf = \frac{1}{4} \bigg( \frac{200\pi}{3} \bigg) \: cm {}^{3} = \frac{50\pi}{3} \: cm {}^{3} \: \: \: \: \: \: \: \: \: \: \: \: \: ...(2)$

$\sf \: Let \: the \: number \: of \: lead \: shots \: dropped \: in \: the \: vessel \: be \: \red{n}.$

$\sf \: Then,$

$\sf \: Volume \: of \: n \: lead \: shots$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \sf = \: n \times volume \: of \: one \: lead \: shot$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \sf = \: \frac{n\pi}{6} \: cm {}^{3} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: [by(1)] \: \: ...(3)$

$\sf \: According \: to \: the \: question, \:$

$\sf \: These \: n \: lead \: shots \: make \: \frac{1}{4} \: of \: the \: water \: \\ \sf \: flow \: out \: of \: the \: cone.$

$\therefore \: \: \: \: \: \sf \: by \: (3) \: and \: (2)$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = \: \sf \frac{n\pi}{6} = \frac{50\pi}{3}$

$\implies \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \sf3n\pi = 300\pi \: \: \: \: \: \: \: \: \: \: \: (Cross - multiplying)$

$\implies \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \sf \: n \: = \frac{300\pi}{3\pi} = 100$

$\sf \: Hence, \: the \: number \: of \: lead \: shots \: dropped \: in \: the \: vessel \: is \: 100.$

Answer 2

★$\huge\underline\bold\green{Given}$★

${Height\: of \: cone\: = \: 8cm}$

${Radius\: of\: cone\: = 5cm}$

${Radius\: of\: lead\: shot\: = 0.5cm}$

$\textbf{one-fourth of the water flows out when}$$\textbf{lead shot dropped in vessel}$

★$\huge\underline\bold\green{Find\: out}$★

$\textbf{Number of lead shot}$

★$\huge\underline\bold\green{Formula\: used}$★

$\textbf{Volume of cone}$

$\large\frac{1}{3}$$\pi{r^2h}$

$\textbf{Volume of sphere}$

$\large\frac{4}{3}$$\pi{r^3}$

★$\huge\underline\bold\green{Solution}$★

$\bold\red{Volume\: of\: cone}$

$\frac{1}{3}$${×}$$\frac{22}{7}$${×}$${5×5×8}$

$\frac{110×40}{21}$

$\frac{4400}{21}$cm³

________________________________

$\bold\red{Volume\: of\: each\: lead\: shot}$

$\large\frac{4}{3}$${×}$$\large\frac{22}{7}$${×}$${0.5×0.5×0.5}$

$\large\frac{2×11×0.5}{21}$

$\large\frac{11}{21}$cm³

________________________

$\bold\red{Volume\: of\: water\: that\: flows\: out\: when}$$\bold\red{lead\: shot\: dropped}$

$\large\frac{1}{4}$${×}$${Volume\: of\: cone}$

$\large\frac{1}{4}$${×}$$\large\frac{4400}{21}$

$\large\frac{1100}{21}$cm³

_______________________________

$\bold\red{Number\: of\: lead\: shot}$

$\large\frac{Volume\: of\: water\: that\: flows\: out}{Volume\: of\: each\: lead\: shot}$

$\large\frac{1100}{21}$${÷}$$\frac{11}{21}$

$\large\frac{1100}{21}$${×}$$\frac{21}{11}$

100 lead shot