Hemispherical Cap Volume Calculation
We need to calculate the volume of a hemispherical cap with a given radius. The question provides the radius $r = 33.6$ cm. Based on the context and the format of the answer options, it is implied that we need to find the volume of a full hemisphere with this radius.
Hemisphere Volume Formula Explained
The volume of a full sphere is given by the formula $V_{sphere} = \frac{4}{3}\pi r^3$.
A hemisphere is exactly half of a sphere. Therefore, the formula for the volume of a hemisphere ($V_{hemisphere}$) is derived by taking half the volume of the sphere:
$$V_{hemisphere} = \frac{1}{2} \times V_{sphere}$$ $$V_{hemisphere} = \frac{1}{2} \times \frac{4}{3}\pi r^3$$ $$V_{hemisphere} = \frac{2}{3}\pi r^3$$In this formula, '$r$' represents the radius of the hemisphere.
Step-by-Step Volume Calculation
We are given the radius $r = 33.6$ cm.
First, we need to calculate the cube of the radius, denoted as $r^3$:
$$r^3 = (33.6)^3$$Let's perform the calculation:
$33.6 \times 33.6 = 1128.96$
$1128.96 \times 33.6 = 37933.056$
So, the value of $r^3$ is $37933.056$ $cm^3$.
Now, we substitute this value into the hemisphere volume formula:
$$V_{hemisphere} = \frac{2}{3}\pi r^3$$Substituting $r^3 = 37933.056$:
$$V_{hemisphere} = \frac{2}{3}\pi (37933.056)$$To find the volume, we multiply $37933.056$ by 2:
$$V_{hemisphere} = \frac{2 \times 37933.056}{3}\pi$$ $$V_{hemisphere} = \frac{75866.112}{3}\pi$$Finally, we divide the result by 3:
$$V_{hemisphere} = 25288.704\pi$$Rounding the Volume Result
The question asks for the volume to be rounded off to two decimal places.
Rounding the calculated volume $25288.704\pi$ $cm^3$ to two decimal places gives:
$$V_{hemisphere} \approx 25288.70\pi$$Final Answer Verification
The calculated volume, rounded to two decimal places, is $25,288.70\pi$ $cm^3$. This result matches the value provided in Option 3.
Important Questions from 3-D Mensuration
- Find the surface area of a sphere whose diameter is equal to 98 cm.
- A conical vessel has base radius 31 cm and height 45 cm. Water is poured into the vessel until it is $\frac{2}{3}$ full. Find the volume (in cm³) of water in the vessel.
- A solid metallic sphere of radius 10 cm is melted and recast into 125 identical spheres. What is the ratio of the surface area of the original sphere to the total surface area of 6 smaller spheres so formed?
- Find the volume (in cm³) of the largest right circular cone that can be cut out from a cube with an edge of 8 cm. Use $\pi = \frac{22}{7}$
- The volume of a solid cylinder is 5852 cm³ and its height is 38 cm. What is the total surface area of the solid cylinder? (Round your answer to the nearest integer) (Use $\pi = \frac{22}{7}$)
