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.
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Important Questions from Mensuration
In a circular garden of radius 15 m, a path of 2 m wide has to be made inside the garden at the rate of ₹ 24 per sq. m. The cost of making the path is: (Take π = \(\frac{22}{7}\) )
A hollow cylinder with outer radius 4 cm and height 2 cm is made up of 1 cm thick metal sheet. What is the volume of metal used? (Take π = \(\frac{22}{7}\))
The sides of a triangular park are 60 m, 297 m and 303 m. Its area is equal to the area of a square-shaped garden. What is the perimeter (in m) of the garden?
The length and breadth of a rectangular field are in the ratio 4 : 3. If the cost of cultivating the field at ₹2 per m 2is ₹600, then the length of the field is:
A cylindrical tank has a capacity of 5632 m3. If the diameter of its base is 8 m, what is the depth of the cylindrical tank? (Use π = \(\frac{22}{7}\))
