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
- 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 ?
- A metallic sheet of rectangular shape is 48 cm × 36 cm. From each corner a square of side x cm is cut off and an open box is made of the remaining sheet. If the box holds 5.12 litre of water, then what is the value of x ?
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 = \(22\over7\)]
Which of the following is a geometrical figure with a three-dimensional geometry that has eight vertices and six rectangular faces?
- 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 $\pi = \frac{22}{7}$
