A solid sphere of radius 3 cm is melted to form a hollow cylinder of height 4 cm and external diameter 10 cm. What is the thickness of the cylinder?
1.00 cm
Calculating Cylinder Thickness from Melted Sphere Volume
This problem involves a change of form from a solid sphere to a hollow cylinder. A key principle in such problems is the conservation of volume. When a solid object is melted and reshaped into a new object, the total volume of material remains the same. Therefore, the volume of the original solid sphere is equal to the volume of the hollow cylinder formed.
Step-by-Step Volume Calculation
1. Volume of the Solid Sphere
The sphere has a radius of 3 cm. The formula for the volume of a sphere is \(V_{sphere} = \frac{4}{3}\pi r^3\), where \(r\) is the radius.
Given radius \(r_{sphere} = 3\) cm.
Volume of sphere \(V_{sphere} = \frac{4}{3}\pi (3 \, \text{cm})^3 = \frac{4}{3}\pi (27 \, \text{cm}^3)\)
\(V_{sphere} = 36\pi \, \text{cm}^3\)
2. Volume of the Hollow Cylinder
The hollow cylinder has a height of 4 cm and an external diameter of 10 cm. We need to find its thickness.
- Height of cylinder \(h = 4\) cm
- External diameter = 10 cm
- External radius \(R = \frac{\text{External diameter}}{2} = \frac{10}{2} = 5\) cm
Let the thickness of the cylinder be \(t\) cm.
The internal radius (\(r\)) of the hollow cylinder is the external radius minus the thickness.
- Internal radius \(r = R - t = (5 - t)\) cm
The volume of a hollow cylinder is the volume of the outer cylinder minus the volume of the inner cylinder. The formula for the volume of a cylinder is \(V_{cylinder} = \pi r^2 h\).
Volume of outer cylinder (using external radius \(R\)) = \(\pi R^2 h = \pi (5 \, \text{cm})^2 (4 \, \text{cm}) = \pi (25 \, \text{cm}^2) (4 \, \text{cm}) = 100\pi \, \text{cm}^3\)
Volume of inner cylinder (using internal radius \(r\)) = \(\pi r^2 h = \pi (5-t \, \text{cm})^2 (4 \, \text{cm}) = 4\pi (5-t)^2 \, \text{cm}^3\)
Volume of hollow cylinder \(V_{cylinder} = \text{Volume of outer cylinder} - \text{Volume of inner cylinder}\)
\(V_{cylinder} = 100\pi - 4\pi (5-t)^2 \, \text{cm}^3\)
3. Equating Volumes and Solving for Thickness
According to the principle of conservation of volume:
\(V_{sphere} = V_{cylinder}\)
\(36\pi = 100\pi - 4\pi (5-t)^2\)
We can divide both sides by \(\pi\):
\(36 = 100 - 4(5-t)^2\)
Now, let's rearrange the equation to solve for \(t\):
\(4(5-t)^2 = 100 - 36\)
\(4(5-t)^2 = 64\)
Divide both sides by 4:
\((5-t)^2 = \frac{64}{4}\)
\((5-t)^2 = 16\)
Take the square root of both sides. Since radius and thickness must be positive, we consider the positive square root:
\(5-t = \sqrt{16}\)
\(5-t = 4\)
Solve for \(t\):
\(t = 5 - 4\)
\(t = 1\) cm
The thickness of the hollow cylinder is 1 cm.
| Object | Property | Value |
|---|---|---|
| Solid Sphere | Radius | 3 cm |
| Hollow Cylinder | Height | 4 cm |
| Hollow Cylinder | External Diameter | 10 cm |
| Hollow Cylinder | External Radius (R) | 5 cm |
| Hollow Cylinder | Internal Radius (r) | 5 - t cm |
| Hollow Cylinder | Thickness (t) | ? |
Comparing the calculated thickness with the given options, we find that 1.00 cm matches one of the choices.
Revision Table - Sphere and Cylinder Volume
| Shape | Key Parameters | Volume Formula |
|---|---|---|
| Sphere | Radius (r) | \(\frac{4}{3}\pi r^3\) |
| Solid Cylinder | Radius (r), Height (h) | \(\pi r^2 h\) |
| Hollow Cylinder | External Radius (R), Internal Radius (r), Height (h) | \(\pi (R^2 - r^2) h\) or \(\pi (R-r)(R+r) h\) |
Note that the formula \(\pi (R^2 - r^2) h\) for a hollow cylinder is equivalent to \(\pi R^2 h - \pi r^2 h\), which is (Volume of outer cylinder) - (Volume of inner cylinder), as used in our step-by-step solution.
Additional Information - Conservation of Volume
The principle of conservation of volume is fundamental in problems where a substance changes shape but not its amount. This applies to melting and recasting metals, reshaping clay, or even changing the container of a liquid (assuming no spills). The volume remains constant regardless of the form it takes. This concept is widely used in geometry and physics problems involving transformations of shapes.
For a hollow cylinder, the thickness \(t\) is the difference between the external radius \(R\) and the internal radius \(r\), i.e., \(t = R - r\). Knowing any two of these values allows you to find the third. In this problem, we used \(r = R - t\) to set up the equation based on volumes.
Similar Questions
A cone and a hemisphere have equal bases and volumes. What is the ratio of the height of the cone to the radius of the hemisphere?
The volume of a hemisphere is 155232 cm 3. What is the radius of the hemisphere?
The radius and height of a right circular cone are in the ratio 3 : 7. If the volume of the cone is 528 cm 3, then what is the height of the cone? \(\left( {{\rm{Take}}\,\,{\rm{\pi }}\,{\rm{ = }}\frac{{22}}{7}} \right)\)
The length, breadth and height of a cuboid are in the ratio 27 : 8 : 1. The cuboid is melted and recast into a cube. If p is the surface area of the cuboid and q is the surface area of the cube, then what is p/q equal to?
A square sheet of side length 44 cm is rolled along one of its sides to form a cylinder by making opposite edges just to touch each other. What is the volume of the cylinder ? (Take π = 22/7)
A lamp shade is in the shape of a part of a cone and its top and bottom ends are circles whose circumferences are respectively 30 cm and 40 cm. The perpendicular distance between the ends is 6 cm. If the cone were to be completed, then how far would its vertex be from the top end?
Three solid lead spheres of radius 6 cm, 8 cm and 10 cm are melted together and recast as a solid sphere. What is the percentage diminution of the surface area as compared to the sum of the surface areas of the three spheres ?
A cylindrical pipe has inner diameter of 14 cm. Water flows through it at a rate of 154 litres per minute. What is the speed of water in km/hr? \(\left( {{\rm{Take}}\,\,{\rm{\pi }}\,{\rm{ = }}\frac{{22}}{7}} \right)\)
What is the radius of the base of the cone ?
A cone of height 16 cm and diameter 14 cm is mounted on a hemisphere of same diameter. What is the volume of the solid thus formed? (take π = 22/7)
Important Questions from Solid Figures
A cone and a hemisphere have equal bases and volumes. What is the ratio of the height of the cone to the radius of the hemisphere?
A metallic solid cuboid of dimensions 36 cm × 18 cm × 12 cm is melted and recast in the form of cubes of side 6 cm. Find the number of cubes so formed.
A solid cylinder has a radius of 9 cm and a height of 25 cm. What is the ratio of its total surface area to its curved surface area?
If the surface area of a sphere is 64 π cm 2, then the volume of the sphere is:
Find the surface area of a sphere of diameter 21 cm. (Use π = \(\frac{{22}}{7}\) )
