A block of marble 5 m x 4 m x 2 m in size is cut into rectangular tiles of 1 m x 0.5m size having thickness of 10 cm. Assuming 10% wastage in cutting, how many tiles will be made?
Marble Block Volume Calculation
First, we need to find the total volume of the marble block. The dimensions of the block are given as 5 m, 4 m, and 2 m.
The formula for the volume of a rectangular block (cuboid) is length × width × height.
So, the volume of the marble block ($V_{block}$) is: $V_{block} = 5 \, \text{m} \times 4 \, \text{m} \times 2 \, \text{m}$ $V_{block} = 40 \, \text{m}^3$
Tile Volume Calculation
Next, we calculate the volume of a single rectangular tile. The dimensions are 1 m × 0.5 m with a thickness of 10 cm.
We must ensure all dimensions are in the same unit. Let's convert the thickness from centimeters to meters: $10 \, \text{cm} = \frac{10}{100} \, \text{m} = 0.1 \, \text{m}$
The volume of one tile ($V_{tile}$) is: $V_{tile} = 1 \, \text{m} \times 0.5 \, \text{m} \times 0.1 \, \text{m}$ $V_{tile} = 0.05 \, \text{m}^3$
Wastage Effect on Volume
The problem states that there is a 10% wastage in cutting. This means that only 90% of the total marble block volume is actually usable for making tiles.
To find the usable volume ($V_{usable}$), we calculate 90% of the total block volume: $V_{usable} = V_{block} \times (100\% - 10\%)$ $V_{usable} = 40 \, \text{m}^3 \times 90\%$ $V_{usable} = 40 \, \text{m}^3 \times \frac{90}{100}$ $V_{usable} = 40 \, \text{m}^3 \times 0.90$ $V_{usable} = 36 \, \text{m}^3$
Number of Tiles Calculation
Finally, to find the total number of tiles that can be made, we divide the usable volume of the marble by the volume of a single tile.
Number of tiles = $\frac{V_{usable}}{V_{tile}}$
Number of tiles = $\frac{36 \, \text{m}^3}{0.05 \, \text{m}^3}$
Number of tiles = $\frac{36}{0.05}$
To simplify the division, we can multiply the numerator and denominator by 100: Number of tiles = $\frac{36 \times 100}{0.05 \times 100} = \frac{3600}{5}$
Number of tiles = 720
Therefore, 720 tiles will be made from the block of marble, considering the 10% wastage.
Important Questions from Mensuration 3D (Notes)
- The height of a cylinder is 14cm and its curved surface area is 264cm². The volume of the cyclinder (in cm³) is:
($\pi=\frac{22}{7}$) - A cylindrical rod has an outer curved surface area of \(7500 \text{ cm}^2\). If the length of the rod is 92 cm, then the outer radius (in cm) of the rod, rounded off to two places of decimal, is:
\(\left(\text{Take } \pi = \frac{22}{7}\right)\) - A number of 512 identical small spheres are cast from a sphere of radius 40 cm, with the total volume of the small spheres being equal to the volume of the larger sphere. The diameter (in cm) of each of the small spheres is:
There is a wooden block in the form of a cube whose each side is 8 meters long.
The maximum possible number of cylinders with a diameter of 1 meter and a height of 4 meters were cut from this block. The cylinders are to be painted at the rate of ₹14 per square meter.
What is the total amount (in ₹) needed to paint all the cylinders if we paint the entire surface of each cylinder? (Take $\pi = \frac{22}{7}$)
- If the lateral surface area of a cylinder is $140.1 \text{ cm}^2$ and its height is $3 \text{ cm}$, then find its volume. (Use $\pi = 3.14$ and round off to two decimal places.)
