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 total surface area of a closed cylinder having a base radius of 70 m and a height of 110 m. [Use π = \(22\over7\)]
- The curved surface area of a cylinder is half of its total surface area. If its height is 195 cm, then find its diameter (in cm).
The diameter of the base and slant height of a right circular cone are 30 cm and 113 cm, respectively. Find the volume (in cm³) of the given cone.
(Use $\pi = \frac{22}{7}$)- A cylindrical rod has an curved surface area of $4,900 \text{ cm}^2$. If the length of the rod is 97 cm, then the radius (in cm) of the rod, correct to two places of decimal, is:
Take $\pi = \frac{22}{7}$ - The volume (in $m^3$) of a cube, each of whose edges is 42 m, is:
