Question

A solid consisting of a right circular cone of height 120 cm and radius 60 cm standing on
a hemisphere of radius 60 cm is placed upright in a right circular cylinder full of water
such that it touches the bottom. Find the volume of water left in the cylinder, if the radius
of the cylinder is 60 cm and its height is 180 cm.

Answer 1

Solution:

Note: Diagram of this question attach in attachment file.

Given:  

$\implies Height\;of\;conical\;part = 120m \\ \\ \implies Radius\;of\;conical\;part=60cm \\ \\ \implies Height\;of\;cylinder = 180 cm$

Formula used:

$\implies Volume\;of\;cone=\dfrac{h}{3} \pi r^{2}h \\ \\ \implies Volume\;of\;hemisphere = \dfrac{2}{3}\pi r^{2} \\ \\ \implies Volume\;of\;cylinder = \pi r^{2}h$

So,

$\implies Volume\;of\;conical\;part=\dfrac{1}{3}\times \dfrac{22}{7}\times 60^{2}\times 120 cm^{3} \\ \\ \\ \implies Radius\;of\;hemisphere\;part=60cm.\\ \\ \implies Volume\;of\;hemisphere\;part = \dfrac{2}{3}\times\dfrac{22}{7}\times 60^{3}$

$\therefore Volume\;of\;solid=[Volume\;of\;conical\;part]+[Volume\;of\;hemisphere\;part]$

$\implies \bigg[\dfrac{1}{3}\times \dfrac{22}{7} \times 60^{2} \times 120\bigg]+\bigg[\dfrac{2}{3}\times \dfrac{22}{7}\times 60^{3}\bigg]$

$\implies \dfrac{2}{3}\times \dfrac{22}{7} \times 60^{2}[60+60]$

$\implies \dfrac{2}{3}\times\dfrac{22}{7}\times 60\times 60\times 120 \\ \\ \implies \dfrac{2\times 22\times 60\times 60\times 40}{7} cm^{3} \\ \\ \implies \dfrac{6336000}{7} cm^{3}$

$\implies volume\;of\;cylinder=\pi r^{2}h \\ \\ \implies \dfrac{22}{7}\times 60^{2}\times 180 \\ \\ \implies \dfrac{22\times 60\times 60\times 180}{7} \\ \\ \implies \dfrac{14256000}{7}cm^{3}$

$\implies volume\;of\;water\;in\;cylinder=\dfrac{14256000}{7}cm^{3}\\ \\ \\ \therefore volume\;of\;water\;left\;in\;the\;cylinder= \Bigg[\dfrac{14256000}{7}-\dfrac{6336000}{7}\Bigg]=\dfrac{7920000}{7}cm^{3}$

$\implies 1131428.57142 cm^{3}\\ \\ \\ \implies we\;convert\;it\;into\;m^{3}\\ \\ \implies \dfrac{1131428.57142}{1000000}m^{3}=1.131 m^{3}$

Answer 2

Answer:

Volume of Water left in Cylinder = 1131428.57 cm³

Step-by-step explanation:

Given :

Height of Cone, h = 120 cm

Radius of Cone, r = 60 cm

Radius of Hemisphere, r = 60 cm

Height of Cylinder, H = 180 cm

Radius of Cylinder, r = 60 cm

To Find :

Volume of water left in cylinder.

Solution :

We know that :

Volume of water left in cylinder :

$\bigstar\;\boxed{\textsf {Volume of cylinder - (Volume of cone + Colume of the hemisphere)}}}$

Values in Equation :

⇒ πr² H - ($\dfrac{1}{3}$ πr²h + $\dfrac{2}{3}$ πr³)

⇒ πr² (H - ([/tex]$\dfrac{1}{3}$ h + $\dfrac{2}{3}$ πr))

⇒ $\dfrac{22}{7}$ × 60² (180 - ($\dfrac{1}{3}$ 120 + $\dfrac{2}{3}$ 60))

⇒ $\dfrac{22}{7}$ × 3600(180 - (40 + 40))

⇒ $\dfrac{22}{7}$ × 3600 × 100

⇒ 1131428.57 cm³

∴ Volume of Water left in Cylinder = 1131428.57 cm³