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 $\pi = \frac{22}{7}$)
Problem Breakdown:
- A wooden cube block has sides of 8 meters.
- We need to cut cylinders with a diameter of 1 meter and a height of 4 meters.
- The goal is to find the maximum number of such cylinders that can be cut.
- Then, calculate the total cost to paint the entire surface of these cylinders at a rate of ₹14 per square meter.
- Use $\pi = \frac{22}{7}$.
1. Calculate Maximum Number of Cylinders
The cube has dimensions 8m x 8m x 8m.
Each cylinder has a diameter of 1m (radius $r = 0.5$m) and a height of 4m.
We need to orient the cylinders to maximize the count.
- Option A: Align cylinder height (4m) along the cube's height (8m).
- Number of layers vertically = $\frac{8\text{m}}{4\text{m}} = 2$ layers.
- On each layer (base), we can fit cylinders with 1m diameter within an 8m x 8m area.
- Number of cylinders per layer = $\frac{8\text{m}}{1\text{m}} \times \frac{8\text{m}}{1\text{m}} = 8 \times 8 = 64$.
- Total cylinders = $2 \times 64 = 128$.
- Option B: Align cylinder diameter (1m) along the cube's height (8m).
- Number of layers vertically = $\frac{8\text{m}}{1\text{m}} = 8$ layers.
- On each layer (base), we can fit cylinders with 4m height within an 8m x 8m area.
- Number of cylinders per layer = $\frac{8\text{m}}{4\text{m}} \times \frac{8\text{m}}{4\text{m}} = 2 \times 2 = 4$.
- Total cylinders = $8 \times 4 = 32$.
The maximum number of cylinders is 128.
2. Calculate Surface Area of One Cylinder
The formula for the Total Surface Area (TSA) of a cylinder is $2 \pi r h + 2 \pi r^2$.
- Radius ($r$) = 0.5 m
- Height ($h$) = 4 m
- $\pi = \frac{22}{7}$
TSA = $2 \times \frac{22}{7} \times 0.5 \times 4 + 2 \times \frac{22}{7} \times (0.5)^2$
TSA = $\frac{2 \times 22 \times 0.5 \times 4}{7} + \frac{2 \times 22 \times 0.25}{7}$
TSA = $\frac{88}{7} + \frac{11}{7}$
TSA = $\frac{99}{7}$ square meters.
3. Calculate Total Painting Cost
Total surface area of all 128 cylinders = $128 \times \frac{99}{7}$ sq m.
Painting rate = ₹14 per square meter.
Total Cost = (Total surface area) $\times$ (Rate per sq m)
Total Cost = $\left( 128 \times \frac{99}{7} \right) \times 14$
Total Cost = $128 \times 99 \times \frac{14}{7}$
Total Cost = $128 \times 99 \times 2$
Total Cost = $128 \times 198$
Total Cost = ₹25,344
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:
- 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.)
- Six cubes each of side 4 cm, are placed adjacent to each other. Find the volume of the cuboid so formed.
