Question

$\underline{\underline{\red{\bf{Question?}}}}$

◕ A vessel full of water is in the form of an inverted cone of height 8 cm and the radius of its top, which is open, is 5 cm. 100 spherical lead balls are dropped into the vessel. one-fourth of the water flows out of the vessel. find the radius of a spherical ball.

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Answer 1

Qᴜᴇsᴛɪᴏɴ :

➥ A vessel full of water is in the form of an inverted cone of height 8 cm and the radius of its top, which is open, is 5 cm. 100 spherical lead balls are dropped into the vessel. one-fourth of the water flows out of the vessel. find the radius of a spherical ball.

Aɴsᴡᴇʀ :

➥ Radius of spherical ball = 0.5 cm

Gɪᴠᴇɴ :

➤ Height of the conical vessel (h) = 8 cm

➤ Radius of the conical vessel (r) = 5 cm

Tᴏ Fɪɴᴅ :

➤ Radius of spherical ball = ?

Sᴏʟᴜᴛɪᴏɴ

Volume of the conical vessel = $\sf{\dfrac{1}{3}}$πr²h

☛ On putting values

$\sf{ :\implies \left( \dfrac{1}{3}\pi \times 5 \times 5 \times 8 \right) {cm}^{3} }$

$\sf{ :\implies \left( \dfrac{200\pi}{3} \right) {cm}^{3} }$

Volume of water flown out on dropping spherical balls

$\sf{ :\implies \left( \dfrac{1}{4} \: of \: \dfrac{200\pi}{3} \right) {cm}^{3} }$

$\sf{ :\implies \left( \dfrac{50\pi}{3} \right) {cm}^{3} }$

Let the radius of each spherical ball be R.

Volume of each spherical ball = $\sf{\dfrac{4}{3}}$πR²

Now, volume of 100 balls = volume of water flow out

$\sf{: \implies 100 \times \dfrac{4}{3}\pi {R}^{2} = \left( \dfrac{50\pi}{3} \right) {cm}^{3} }$

$\sf{: \implies {R}^{3} = \left( \dfrac{50}{3} \times \dfrac{3}{4} \times \dfrac{1}{100} \right) {cm}^{3} }$

$\sf{: \implies {R}^{3} = \dfrac{1}{8} \: {cm}^{3} }$

$\sf{: \implies R= \sqrt[3]{ \dfrac{1}{8} } \: cm}$

$\sf{: \implies R=\dfrac{1}{2} \: cm}$

$\sf{: \implies \underline{ \overline{ \boxed{ \purple{ \bf{ \: \: R= 0.5 \: cm \: \: }}}}}}$

Hence, the radius of a spherical ball is 0.5 cm.

Answer 2

$\Huge{\purple{\underline{\textsf{Solution}}}}$

★${\red{\underline{\bf{Volume\:of\:water\:in\:cone\::-}}}}$

$\implies \: \pink{\sf \: \dfrac{1}{3} \pi \: r^{2} h} \\ \\ \\ \implies \: \pink{\sf \: \dfrac{1}{3} \pi \: (5 ) ^{2} h} \\ \\ \\ \implies \: \pink{\sf \: \dfrac{200}{3} \pi \: cm^{3} }$

★${\red{\underline{\bf{Volume\:of\:water\:flows\:out\::-}}}}$

$\implies \: \orange{\sf \: \dfrac{1}{4} \times \:\dfrac{200}{3}\pi} \\ \\ \\ \: \implies \: \orange{\sf \:\dfrac{50}{3}\pi \: cm^{3} }$

$\bigstar\:{\red{\underline{\bf{Let\:the\:radius\:of\:one \: \: spherical \: ball\:be\:r\:cm:-}}}}$

$\longrightarrow\:$$\gray{\sf \dfrac{4}{3}\pi r^3 \times 100 = \dfrac{50}{3} \pi }$

$\longrightarrow\:$$\gray{\sf r^3 = \dfrac{50}{4 \times 100}}$

$\longrightarrow\:$$\gray{\sf r^3 = \dfrac{1}{8}}$

$\longrightarrow\:$$\gray{\sf r = \dfrac{1}{2}}$

$\longrightarrow\:$$\underline{\boxed{\gray{\bf r = 0.5\:cm}}}$$\green\bigstar$

Therefore, the radius of spherical ball is 0.5 cm.