A solid metallic rectangular block of dimensions 112 cm × 44 cm × 25 cm is melted and recast into a cylinder of radius 35 cm. The curved surface area (in cm 2) of the cylinder is: (Take π = 22/7)
7040
Finding Cylinder Curved Surface Area from Melted Block
This problem involves the principle of conservation of volume. When a solid metallic rectangular block is melted and recast into a cylinder, the volume of the material remains the same. Therefore, the volume of the rectangular block is equal to the volume of the cylinder.
Step 1: Calculate the Volume of the Rectangular Block
The dimensions of the rectangular block are given as length (l) = 112 cm, breadth (b) = 44 cm, and height (h) = 25 cm.
The formula for the volume of a rectangular block (cuboid) is:
\( \text{Volume} = \text{Length} \times \text{Breadth} \times \text{Height} \)
Substituting the given values:
\( \text{Volume of Block} = 112 \, \text{cm} \times 44 \, \text{cm} \times 25 \, \text{cm} \)
\( \text{Volume of Block} = 112 \, \text{cm} \times (44 \times 25) \, \text{cm}^2 \)
\( \text{Volume of Block} = 112 \, \text{cm} \times 1100 \, \text{cm}^2 \)
\( \text{Volume of Block} = 123200 \, \text{cm}^3 \)
So, the volume of the rectangular block is \(123200 \, \text{cm}^3\).
Step 2: Equate Volumes to Find Cylinder Height
Since the rectangular block is melted and recast into a cylinder, the volume of the cylinder is equal to the volume of the block.
\( \text{Volume of Cylinder} = \text{Volume of Block} = 123200 \, \text{cm}^3 \)
The formula for the volume of a cylinder is:
\( \text{Volume} = \pi r^2 h \)
where \(r\) is the radius and \(h\) is the height of the cylinder.
We are given the radius of the cylinder, \(r = 35 \, \text{cm}\), and we need to find the height, \(h\).
Using the value of \(\pi = \frac{22}{7}\):
\( 123200 = \frac{22}{7} \times (35)^2 \times h \)
\( 123200 = \frac{22}{7} \times (35 \times 35) \times h \)
\( 123200 = \frac{22}{7} \times 1225 \times h \)
Simplify the term with \(\frac{22}{7}\) and 1225:
\( \frac{22}{7} \times 1225 = 22 \times \frac{1225}{7} = 22 \times 175 = 3850 \)
So the equation becomes:
\( 123200 = 3850 \times h \)
Now, solve for \(h\):
\( h = \frac{123200}{3850} \)
\( h = \frac{12320}{385} \)
To simplify the fraction, we can divide both numerator and denominator by common factors. Both are divisible by 5:
\( h = \frac{12320 \div 5}{385 \div 5} = \frac{2464}{77} \)
Now, divide 2464 by 77. \(77 \times 3 = 231\), \(246 - 231 = 15\). Bring down 4, making 154. \(77 \times 2 = 154\).
\( h = \frac{2464}{77} = 32 \)
The height of the cylinder is \(32 \, \text{cm}\).
Step 3: Calculate the Curved Surface Area of the Cylinder
The formula for the curved surface area (CSA) of a cylinder is:
\( \text{CSA} = 2 \pi r h \)
We have \(r = 35 \, \text{cm}\), \(h = 32 \, \text{cm}\), and \(\pi = \frac{22}{7}\).
Substitute these values into the formula:
\( \text{CSA} = 2 \times \frac{22}{7} \times 35 \, \text{cm} \times 32 \, \text{cm} \)
\( \text{CSA} = (2 \times \frac{22}{7} \times 35) \times 32 \, \text{cm}^2 \)
Simplify the terms in the parenthesis:
\( 2 \times \frac{22}{7} \times 35 = 2 \times 22 \times \frac{35}{7} = 2 \times 22 \times 5 = 44 \times 5 = 220 \)
So the CSA calculation becomes:
\( \text{CSA} = 220 \times 32 \, \text{cm}^2 \)
\( \text{CSA} = 7040 \, \text{cm}^2 \)
The curved surface area of the cylinder is \(7040 \, \text{cm}^2\).
Let's summarise the steps and results:
| Step | Calculation | Result |
|---|---|---|
| 1 | Volume of Rectangular Block = \(112 \times 44 \times 25\) | \(123200 \, \text{cm}^3\) |
| 2 | Volume of Cylinder = Volume of Block \( \frac{22}{7} \times 35^2 \times h = 123200 \) |
\( h = 32 \, \text{cm} \) |
| 3 | Curved Surface Area of Cylinder = \( 2 \pi r h \) \( 2 \times \frac{22}{7} \times 35 \times 32 \) |
\( 7040 \, \text{cm}^2 \) |
Final Answer Check
The calculated curved surface area of the cylinder is 7040 cm\(^2\).
Revision Table: Volume and Surface Area Formulas
| Shape | Volume Formula | Curved Surface Area (CSA) Formula |
|---|---|---|
| Rectangular Block (Cuboid) | \( \text{Length} \times \text{Breadth} \times \text{Height} \) | \( 2(\text{Length} + \text{Breadth}) \times \text{Height} \) (Lateral Surface Area) |
| Cylinder | \( \pi r^2 h \) | \( 2 \pi r h \) |
Additional Information: Conservation of Volume
The principle of conservation of volume is fundamental in problems where a substance is reshaped. When a solid is melted and recast, or molded into a different shape, the total amount of material, and hence its volume, remains unchanged. This is a crucial concept in mensuration and solid geometry problems.
In this problem, the metallic material from the rectangular block is simply changing its form from a block shape to a cylindrical shape. No material is added or removed during the melting and recasting process, which is why the volume stays constant.
Important Questions from Solid Figures
The area of the floor of a cubical room is 192 m 2. The length of the longest rod that can be kept in that room is :
If the volume of a cube is 175616 cm 3, what is its side?
The volume of a right circular cone is 1232 cm 3. If the height of the cone is 24 cm, then what will be the radius of its base?
A right triangle contains the right angle between the sides 5 cm and 7 cm. A cone is generated by revolving about the side 5 cm. The volume of this cone is:
The area of the base of a right circular cone is \(\frac{1408}{7} cm^2\) and its height is 6 cm. Taking π = \(\frac{22}{7}\) , the curved surface area of the cone is:
