Question

A cone, a hemisphere and a cylinder stand on equal basis and have the same height. Find the
ratio of their volumcs.​

Answer 1

$\bigstar \underline{\underline{\bf Given:-}}$

• A cone, a hemisphere and a cylinder stand on equal basis
• They have the same height.

$\bigstar \underline{\underline{\bf To\ find:-}}$

• The ratio of their volumes.

$\bigstar \underline{\underline{\bf Solution:-}}$

We need to know:

❥ Volume of cone = (1/3)πr²h

❥Volume of hemisphere = (2/3)πr³ h

❥Volume of cylinder = πr²h

----------------------------

Let radius = r & height = h,

Given that A cone ,a hemisphere & a cylinder stand on equal basis .So,

=> Radius of base of cone = radius of hemisphere = radius of cylinder.

=>Height of cone = height of hemisphere = height of cylinder

→Height of Hemisphere r=h

Now,

➲ Volume of cone = (1/3)πr²h

➲Volume of Hemisphere = (2/3)πr³

$:\implies \sf \frac{2}{3} \pi r^2 \times (r)\\\\:\implies \sf \frac{2}{3} \pi r^2 \times (h)\ \ [r=h] \\\\:\implies \sf V_H = \sf \frac{2}{3} \pi r^2 h$

➲Volume of cylinder=πr²h

Ratio of their volumes :-

➤Vol of cone: Vol of hemisphere:Vol of cylinder.

$:\implies \sf \frac{1}{3} \pi r^2 h:\frac{2}{3} \pi r^2 h : \pi r^2 h$

Cancelling out r²,π,h as they are in common.

$:\implies \sf \frac{1}{3} :\frac{2}{3} :1\\\\:\implies \boxed{\boxed{\sf 1:2:3}}$

Therefore,

The ratio of their volumes is 1:2:3.

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Answer 2

Answer:

$\huge\rm{Question}$

A cone, a hemisphere and a cylinder stand on equal bases and have the same height. Show that their volumes are in the ratio 1:2:3.

$\huge\rm{Solution:–}$

$\rm \: r=h \\ \rm \: Vc:Vh:Vcyl \\ \rm \: → \frac{1}{3} \pi \: r².h= \frac{2}{3} \pi \: r³:\pi \: r²h \\ \rm \: → \frac{1}{3} \pi \: r³: \frac{2}{3} \pi \: r³:\pi \: r³ \\ \rm \: →\pi \: r³:2πr³:3πr³ \\ \rm \: →1:2:3 \\ \\ \rm \: So \: the \: required \: answer \: is \: {\underline{\underline{1:2:3}}}$