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GO Classes | IIITH PGEE 2026 | Memory Based | Question: 14

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A solid cylinder has a height of $15\mathrm{cm}$ and a diameter of $10\mathrm{cm}$. From the top of the cylinder, a conical cavity of the same base diameter and height $6\mathrm{cm}$ is removed. At the bottom of the cylinder, a hemisphere of the same diameter is attached. Find the total volume of the resulting solid.

  1. $\frac{1125\pi}{3}\mathrm{cm}^3$
     
  2. $\frac{1175\pi}{3}\mathrm{cm}^3$
     
  3. $\frac{1225\pi}{3}\mathrm{cm}^3$
     
  4. $\frac{1275\pi}{3}\mathrm{cm}^3$

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Answer:C

\[
\text{Radius of the cylinder} = \frac{10}{2} = 5 \text{ cm}
\]

\[
\text{Height of the cylinder} = 15 \text{ cm}
\]

Volume of the cylinder:

\[
V_{\text{cyl}} = \pi r^2 h
= \pi (5)^2(15)
= 375\pi
\]

A conical cavity is removed with:

\[
r = 5 \text{ cm}, \qquad h = 6 \text{ cm}
\]

Volume of the cone removed:

\[
V_{\text{cone}} = \frac{1}{3}\pi r^2 h
= \frac{1}{3}\pi (5)^2(6)
= 50\pi
\]

A hemisphere is attached with radius:

\[
r = 5 \text{ cm}
\]

Volume of the hemisphere:

\[
V_{\text{hemi}} = \frac{2}{3}\pi r^3
= \frac{2}{3}\pi (5)^3
= \frac{250\pi}{3}
\]

Total volume of the resulting solid:

\[
V
= 375\pi - 50\pi + \frac{250\pi}{3}
\]

\[
= 325\pi + \frac{250\pi}{3}
\]

\[
= \frac{975\pi + 250\pi}{3}
\]

\[
= \frac{1225\pi}{3}
\]

\[
\boxed{\frac{1225\pi}{3}\ \text{cm}^3}
\]
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