Question

the radii of the base of a cylinder and a cone are in the ratio √3:√2 and their Heights are in the ratio √2:√3. Find the ratio of their volumes.

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Answer 1

Required Answer:-

Given the ratio of radius and heights of a cylinder and a cone

• Ratio of radius = √3 : √2
• Ratio of heights = √2 : √3

Assuming the Radius be √3r & √2r since they are in a ratio and the heights be √2h & √3h are also in ratio. Now we know the formula for volume of a cylinder.

➙ $\underline{ \boxed{ \large{ \bf{v = \pi {r}^{2} h}}}}$

And the volume of cone:

➙ $\underline{ \boxed{ \large{ \bf{v = \frac{1}{3} \pi {r}^{2} h}}}}$

Plugging the assumed values of radius and heights to find their volume ratios:

➙ $\rm{ \dfrac{v1}{v2} = \dfrac{\pi( \sqrt{3}r) {}^{2} \sqrt{2}h }{1/3 \pi( \sqrt{2}) {}^{2} \sqrt{3} h } }$

This would be equals to:

➙ $\rm{ \dfrac{v1}{v2} = \dfrac{\pi \times 3 {r}^{2} \times \sqrt{2} h}{1/3\pi \times 2 {r}^{2} \times \sqrt{3} h } }$

Now taking πr²h common from both numerator and denominator:

➙ $\rm{ \dfrac{v1}{v2} = \dfrac{3 \sqrt{2}(\pi {r}^{2} h) }{2 \sqrt{3}/3 (\pi {r}^{2}h) } }$

πr²h will be cancelled. Hence our volume ratio will remain 3√2 : 2√3/3. Simplifying further.

➙ Ratio of their volumes: $\rm{ \boxed{ \red{3 \sqrt{3} : \sqrt{2} }}}$

Answer 2

Answer:

Required Answer :-

Let us assume the height of two circle as √2h and √3h

and radius be √3r and √2r

As we know that

$\bf \: V = \bold{\pi} r {}^{2} h$

Now,

Put assumed height and radius

$\tt \: \dfrac{v1}{v2} = \dfrac{\pi \sqrt{3r} \sqrt{2h} }{\pi \sqrt{2r} \sqrt{3h} }$

Now,

$\tt \dfrac{v1}{v2} = \dfrac{\pi \times {9}^{2} \times \sqrt{2h} }{\pi \times {4}^{2} \times \sqrt{3h} }$

$\tt \: \dfrac{v1}{v2} = \dfrac{9 \sqrt{2} \cancel{\pi \: {r}^{2} h}}{4 \sqrt{3} \cancel{ \pi \: {r}^{2}h } }$

Now,

$\tt \: \dfrac{v1}{v2} = \dfrac{9 \sqrt{2} }{4 \sqrt{3} }$

$\frak \pink{Ratio = 3 \sqrt{3} \ratio \: \sqrt{2} }$