Calculating Number of Cones from Melted Sphere
The problem requires finding how many small cones can be made by melting a larger sphere. This involves comparing the volumes of the sphere and the cones.
Sphere Dimensions and Volume
Given the sphere's diameter is 24 cm, its radius (R) is half of that:
Radius of sphere, \( R = \frac{24 \text{ cm}}{2} = 12 \text{ cm} \)
The volume of a sphere is calculated using the formula \( V_{sphere} = \frac{4}{3}\pi R^3 \).
Substituting the radius:
\( V_{sphere} = \frac{4}{3}\pi (12 \text{ cm})^3 \)
\( V_{sphere} = \frac{4}{3}\pi (1728 \text{ cm}^3) \)
\( V_{sphere} = 4 \times \pi \times 576 \text{ cm}^3 \)
\( V_{sphere} = 2304\pi \text{ cm}^3 \)
Cone Dimensions and Volume
Each small cone has a base radius (r) of 4 cm and a height (h) of 9 cm.
The volume of a cone is calculated using the formula \( V_{cone} = \frac{1}{3}\pi r^2 h \).
Substituting the given values:
\( V_{cone} = \frac{1}{3}\pi (4 \text{ cm})^2 (9 \text{ cm}) \)
\( V_{cone} = \frac{1}{3}\pi (16 \text{ cm}^2) (9 \text{ cm}) \)
\( V_{cone} = \pi \times 16 \text{ cm}^2 \times 3 \text{ cm} \)
\( V_{cone} = 48\pi \text{ cm}^3 \)
Number of Cones Calculation
To find the total number of cones obtained, divide the volume of the sphere by the volume of one cone.
Number of cones \( N = \frac{V_{sphere}}{V_{cone}} \)
\( N = \frac{2304\pi \text{ cm}^3}{48\pi \text{ cm}^3} \)
\( N = \frac{2304}{48} \)
\( N = 48 \)
Therefore, 48 cones will be obtained.
Similar Questions
- The volume of a wall that is 5 times as high as it is broad and 8 times as long as it is high is 12.8 cubic meters. Find the breadth of the wall.
- If the radius of a sphere is increased by 2 cm, then its surface area increases by 704 cm\(^2\). Using \(\pi = \frac{22}{7}\), find the radius of the sphere after the increase.
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- Find the area of a regular hexagon whose side measures \(18\sqrt{2}\) cm. (Use \(\sqrt{3} = 1.73\))
The sides of the rectangle are in the ratio 8 : 10 and its perimeter is 90 cm. The area of this rectangle is:
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Important Questions from Mensuration
- The volume of a wall that is 5 times as high as it is broad and 8 times as long as it is high is 12.8 cubic meters. Find the breadth of the wall.
- If the radius of a sphere is increased by 2 cm, then its surface area increases by 704 cm\(^2\). Using \(\pi = \frac{22}{7}\), find the radius of the sphere after the increase.
- What is the perimeter of an equilateral triangle with side 13 cm?
- The perimeter of a square is 76 m. Its area (in $m^2$) is:
- A cylindrical rod has an outer curved surface area of 1800 cm$^2$. If the length of the rod is 52 cm, then the outer radius (in cm) of the rod, correct to two places of decimal, is:
Take $\pi = \frac{22}{7}$
