Calculating Wire Length from Melted Sphere Volume
The core principle is that the volume of the material remains constant when a solid is melted and recast. Therefore, the volume of the copper sphere is equal to the volume of the wire (which is a cylinder).
Sphere Calculations
- Given sphere diameter = 6 cm.
- Sphere radius (\(r_s\)) = Diameter / 2 = 6 cm / 2 = 3 cm.
- Volume of a sphere (\(V_s\)) is given by the formula:
\(V_s = \frac{4}{3} \pi r_s^3\)
- Substituting the radius:
\(V_s = \frac{4}{3} \pi (3 \text{ cm})^3 = \frac{4}{3} \pi (27 \text{ cm}^3) = 36 \pi \text{ cm}^3\)
Wire Calculations
- Given wire diameter = 0.5 cm.
- Wire radius (\(r_w\)) = Diameter / 2 = 0.5 cm / 2 = 0.25 cm.
- The wire is cylindrical. Volume of a cylinder (\(V_w\)) is given by:
\(V_w = \pi r_w^2 h\)
where '\(h\)' is the length of the wire. - Substituting the radius:
\(V_w = \pi (0.25 \text{ cm})^2 h = \pi (0.0625 \text{ cm}^2) h\)
Equating Volumes and Finding Length
- Set the volume of the sphere equal to the volume of the wire:
\(V_s = V_w\)
\(36 \pi \text{ cm}^3 = \pi (0.0625 \text{ cm}^2) h\)
- Cancel \(\pi\) from both sides:
\(36 \text{ cm}^3 = 0.0625 \text{ cm}^2 \times h\)
- Solve for '\(h\)':
\(h = \frac{36 \text{ cm}^3}{0.0625 \text{ cm}^2}\)
\(h = 576 \text{ cm}\)
- Convert the length from centimeters to meters (since 1 m = 100 cm):
\(h = \frac{576}{100} \text{ m} = 5.76 \text{ m}\)
The length of the wire is 5.76 m.
Similar Questions
A room is L m long, B m wide and H m high. Further, L > B > H and L, B and H are integers. The length of the longest pole that can be placed in the room is 17 m and the length of the longest pole that can be placed on the floor is 15 m. What is the volume of the room?
A reservoir is in the form of a cuboid. Its length is 30 m. If 24 kL of water is removed from the reservoir, the water level goes down by 20 cm. What is the width of the reservoir?
A well is dug with a diameter of 3.5 m and 16 m depth. The earth soil so excavated is spread in the form of a right circular cone of radius 7 m. What is the height of the cone?
A tall cylindrical reservoir kept vertically is 20 m in diameter. Water is poured into it at the rate of 264 m³ per hour. What is the rate at which the water level rises in the reservoir per minute? (Take π = 22/7)
A thin metallic sheet, 1.92 m² in area, is cut into two equal pieces. One piece is used to make a hollow cube of volume P (in m³) and the other is used to make a hollow cuboid of volume Q (in m³) with dimensions in the ratio \(4:2:1\). If no sheet is wasted, which one of the following is correct?
How many square metres of canvas (approximately) will be required to make a conical tent 3 m high so that a man 2 m tall may stand anywhere within a radius of 1 m from its centre without stooping?
N number of cubes each of side length equal to 10 cm are joined end to end in a row. If the total surface area of the resulting cuboid is \(3800\text{ cm}^2\), then what is the value of N?
A spherical metal ball is molten and made into n smaller identical spheres. In this process, the surface area of the smaller balls increases by 900%. What is the value of n?
- Let x be the length of a diagonal of a face of a cube and y be the length of a diagonal of the cube. If x + y = \((5 + 2\sqrt{6})\) units, then what is the total surface area of the cube ?
- A sphere of diameter 14 cm is dropped into a cylindrical vessel partly filled with water. The radius of the vessel is 14 cm. If the sphere is completely submerged in water, then how much will the level of water rise ?
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:
