Question

A sphere of diameter 12 cm is dropped in a right circular cylinder , partially filled with water . If the sphere is completely submerged in water the water level rises by 32 / 9 cm .Find the diameter of the cylindrical vessel?

Answer 1

$Diameter \:of \: a \:sphere (d) = 12 \:cm$

$Radius \:of \:the \:sphere (R) = \frac{d}{2} \\= \frac{12}{2} \\= 6 \:cm$

/* According to the problem given */

If the sphere is dropped in a circular cylinder completely submerged in water the water level rises (32/9) cm .

$Height \:of \:water \:level (h) = \frac{32}{9} \:cm$

$Let \: the \:radius \:of \:the \: cylinder = r \:cm$

$Volume_{(Cylinder)} = Volume_{(sphere)}$

$\implies \pi r^{2}h = \frac{4}{3} \pi R^{3}$

$\implies r^{2} \times \frac{32}{9} = \frac{4}{3}\times 6^{3}$

$\implies r^{2} = \frac{4}{3} \times 6 \times 6 \times 6 \times \frac{9}{32} \\= 9 \times 9$

$\implies r = \sqrt{9\times 9}$

$\implies r = 9 \:cm$

$\implies Diameter \:of \;the \: cylindrical \:vessel \\= 2r\\= 2 \times 9 \\= 18 \:cm$

Therefore.,

$\red{Diameter \:of \;the \: cylindrical \:vessel} \\\green {= 18 \:cm }$

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