Question

A conical vessel of radius 6 cm and height 8 cm is completely filled with water. A sphere is lowered in the water and its size is such that when it touches the sides, it is just immersed. What fraction of the water overflows? [4 MARKS]

Answer 1

Radius of conical vessel, R = 6 cm and height of conical vessel, H = 8 cm
$\begin{array}{ccc}Inright-angled\Delta BAC,Wehave& B{C}^2=A{C}^2+A{B}^2& [\because UsingPythagorasTheorem]\\ \Rightarrow & B{C}^2=(8{)}^2+(6{)}^2& \\ \Rightarrow & B{C}^2=64+36=100& \\ \Rightarrow & BC=10cm& \end{array}$
Let radius of the sphere be r cm. Then OA = r cm

$\begin{array}{ccc}\Rightarrow & OC=AC-OA=(8-r)cm& \\ Also,& Angle1={90}^{\circ }& \left[Radiusisperpendiculartothetangent\right]\\ and& AB=BD=6cm& \left[Tangentsdrawnfromexternal\right]\\ & & \left[pointtoacircleareequal\right]\\ Inright-angled\Delta OCD,& O{C}^2=O{D}^2+C{D}^2& [\because UsingPythagorasTheorem]\\ \Rightarrow & (8-r{)}^2=r^2+4^2& [\because UsingPythagorasTheorem]\\ \Rightarrow & 64+r^2-16r=r^2+16& [\because CD=BC-BD=10-6=4cm]\\ \Rightarrow & 64+r^2-16r=r^2+16& \\ \Rightarrow & 48=16r& \\ \Rightarrow & r=3cm& \end{array}$
$\therefore$ Fractionof water which overflows$=\frac{Volumeofsphere}{Totalvolumeofwaterinconebeforeimmersion}=\frac{\frac{4}{3}\pi r^3}{\frac{1}{3}\pi R^{2}H}=\frac{4\times (3{)}^3}{6\times 6\times 8}=\frac{3}{8}$