Two cylindrical vessels are filled with milk. The radius of one vessel is $15 c m$ and height is $40 c m$ , and the radius of other vessel is $20 c m$ and height is $45 c m$ , Find the radius of another cylindrical vessel of height $30 c m$ which may just contain the milk which is in the two given vessels.

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Solution

Given,
Radius of one cylinder $(r_{1}) = 15 c m$
and height $(h_{1}) = 40 c m$
Radius of second cylinder $(r_{2}) = 20 c m$
and height $(h_{2}) = 45 c m$

Now
Volume of first cylinder $π r_{1}^{2} h_{1} = \frac{22}{7} \times 15 \times 15 \times 40 {c m}^{3} = \frac{198000}{7} {c m}^{3}$

Volume of second cylinder $= π r_{2}^{2} h_{2} = \frac{22}{7} \times 20 \times 20 \times 45 = \frac{39600}{7} {c m}^{3}$

So, total volume $= (\frac{198000}{7} + \frac{396000}{7}) {c m}^{3} = \frac{594000}{7} {c m}^{3}$

Now, height $= 30 c m$
Thus,
Radius $= \sqrt[3]{\frac{594000}{7} \times \frac{7}{22} \times \frac{1}{30}} = \sqrt{900} = 30 c m$

Hence, Radius of the third cylinder $= 30 c m$

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Two cylindrical vessels are filled with milk. The radius of one vessel is 15cm and height is 40cm, and the radius of other vessel is 20cm and height is 45cm, Find the radius of another cylindrical vessel of height 30cm which may just contain the milk which is in the two given vessels.