Q7:

A solid consisting of a right circular cone of height $120$ cm and radius $60$ cm standing on a hemisphere of radius $60$ cm is placed upright in a right circular cylinder full of water such that it touches the bottom. Find the volume of water left in the cylinder, if the radius of the cylinder is $60$ cm and its height is $180$ cm.

Given:

• A solid object consisting of a right circular cone mounted on a hemisphere.
• Height of the cone ($h_{cone}$) = $120$ cm.
• Radius of the cone ($r$) = $60$ cm.
• Radius of the hemisphere ($r$) = $60$ cm.
• A right circular cylinder containing water.
• Radius of the cylinder ($R$) = $60$ cm.
• Height of the cylinder ($H$) = $180$ cm.

To Find:

The volume of water left in the cylinder after the solid is submerged in it.

Step 1: Calculate the volume of the cylinder.

The formula for the volume of a cylinder is $V_{cyl} = \pi R^2 H$.

$V_{cyl} = \pi \times (60)^2 \times 180$

$V_{cyl} = \pi \times 3600 \times 180 = 648,000\pi \text{ cm}^3$.

Step 2: Calculate the volume of the solid.

The solid consists of a cone and a hemisphere. The total volume $V_{solid} = V_{cone} + V_{hemisphere}$.

Formula for volume of a cone: $V_{cone} = \frac{1}{3}\pi r^2 h_{cone}$.

$V_{cone} = \frac{1}{3} \times \pi \times (60)^2 \times 120 = \pi \times 3600 \times 40 = 144,000\pi \text{ cm}^3$.

Formula for volume of a hemisphere: $V_{hemisphere} = \frac{2}{3}\pi r^3$.

$V_{hemisphere} = \frac{2}{3} \times \pi \times (60)^3 = \frac{2}{3} \times \pi \times 216,000 = 2 \times \pi \times 72,000 = 144,000\pi \text{ cm}^3$.

Total volume of the solid $V_{solid} = 144,000\pi + 144,000\pi = 288,000\pi \text{ cm}^3$.

Step 3: Calculate the volume of water left in the cylinder.

When the solid is placed in the cylinder, it displaces a volume of water equal to its own volume. The volume of water left is the difference between the volume of the cylinder and the volume of the solid.

$V_{left} = V_{cyl} - V_{solid}$

$V_{left} = 648,000\pi - 288,000\pi = 360,000\pi \text{ cm}^3$.

Step 4: Convert to numerical value using $\pi \approx \frac{22}{7}$.

$V_{left} = 360,000 \times \frac{22}{7} \approx 360,000 \times 3.14159 \approx 1,131,428.57 \text{ cm}^3$.

Converting to cubic meters ($1 \text{ m}^3 = 1,000,000 \text{ cm}^3$):

$V_{left} \approx 1.131 \text{ m}^3$.

Final Answer: The volume of water left in the cylinder is $360,000\pi \text{ cm}^3$ or approximately $1.131 \text{ m}^3$.