A cylindrical tank has a capacity of 5632 m3. If the diameter of its base is 8 m, what is the depth of the cylindrical tank? (Use π = \(\frac{22}{7}\))
112 m
Calculate Depth of a Cylindrical Tank using Volume and Diameter
This problem asks us to find the depth, which is the height, of a cylindrical tank given its capacity (volume) and the diameter of its base. We will use the formula for the volume of a cylinder to solve this.
The volume of a cylinder is given by the formula:
\(V = \pi r^2 h\)
where:
- \(V\) is the volume of the cylinder
- \(\pi\) is a mathematical constant (given as \(\frac{22}{7}\))
- \(r\) is the radius of the base of the cylinder
- \(h\) is the height (or depth) of the cylinder
We are given the following information:
- Volume (\(V\)) = 5632 m\({}^3\)
- Diameter (\(d\)) of the base = 8 m
- Use \(\pi = \frac{22}{7}\)
First, we need to find the radius (\(r\)) from the given diameter (\(d\)). The radius is half of the diameter.
\(r = \frac{d}{2}\)
Substituting the given diameter:
\(r = \frac{8 \text{ m}}{2} = 4 \text{ m}\)
Now we can substitute the known values into the volume formula and solve for the height (\(h\)), which represents the depth of the tank.
\(V = \pi r^2 h\)
\(5632 = \frac{22}{7} \times (4)^2 \times h\)
\(5632 = \frac{22}{7} \times 16 \times h\)
\(5632 = \frac{22 \times 16}{7} \times h\)
\(5632 = \frac{352}{7} \times h\)
To find \(h\), we rearrange the equation:
\(h = \frac{5632 \times 7}{352}\)
Now, we calculate the value of \(h\):
\(h = \frac{39424}{352}\)
Performing the division:
\(39424 \div 352 = 112\)
So, the depth of the cylindrical tank is 112 meters.
Let's verify the calculation:
\(V = \pi r^2 h = \frac{22}{7} \times (4)^2 \times 112 = \frac{22}{7} \times 16 \times 112\)
\(V = \frac{22 \times 16 \times 112}{7} = \frac{22 \times 1792}{7} = \frac{39424}{7}\)
\(39424 \div 7 = 5632\)
The calculated volume matches the given volume, so the depth is correct.
The depth of the cylindrical tank is 112 m.
| Given | Value |
|---|---|
| Volume (V) | 5632 m\({}^3\) |
| Diameter (d) | 8 m |
| \(\pi\) | \(\frac{22}{7}\) |
| Calculation Steps | Result |
|---|---|
| Radius (r = d/2) | 4 m |
| Formula: \(V = \pi r^2 h\) | 5632 = \(\frac{22}{7} \times 4^2 \times h\) |
| Solve for h | \(h = \frac{5632 \times 7}{352}\) |
| Depth (h) | 112 m |
Revision Table: Cylinder Geometry Formulas
| Concept | Formula |
|---|---|
| Area of base (circle) | \(A = \pi r^2\) |
| Circumference of base (circle) | \(C = 2\pi r\) or \(C = \pi d\) |
| Volume of Cylinder | \(V = \pi r^2 h\) |
| Lateral Surface Area of Cylinder | \(LSA = 2\pi r h\) |
| Total Surface Area of Cylinder | \(TSA = 2\pi r (r + h)\) |
Additional Information on Cylinders and Volume
A cylinder is a three-dimensional solid shape that has two parallel circular bases connected by a curved surface. The distance between the two bases is called the height or depth of the cylinder.
The volume of a cylinder essentially measures the amount of space it occupies or the amount of substance it can hold (its capacity). It is calculated by multiplying the area of its circular base by its height.
In this problem, the term 'capacity' is used interchangeably with 'volume', which is common when dealing with containers like tanks. The units for volume are cubic units (m\({}^3\), cm\({}^3\), etc.), while the units for diameter, radius, and height are linear units (m, cm, etc.).
Understanding the relationship between diameter, radius, and height is crucial for solving problems involving cylindrical shapes. Always remember that the radius is half of the diameter.
Using the correct value for \(\pi\) as specified in the question (\(\frac{22}{7}\) in this case) is important for getting the exact answer required.
Important Questions from Mensuration
In a circular garden of radius 15 m, a path of 2 m wide has to be made inside the garden at the rate of ₹ 24 per sq. m. The cost of making the path is: (Take π = \(\frac{22}{7}\) )
A hollow cylinder with outer radius 4 cm and height 2 cm is made up of 1 cm thick metal sheet. What is the volume of metal used? (Take π = \(\frac{22}{7}\))
The sides of a triangular park are 60 m, 297 m and 303 m. Its area is equal to the area of a square-shaped garden. What is the perimeter (in m) of the garden?
The length and breadth of a rectangular field are in the ratio 4 : 3. If the cost of cultivating the field at ₹2 per m 2is ₹600, then the length of the field is:
Find the surface area of a sphere whose diameter is equal to 28 cm.
