The areas of three adjacent faces of a cuboidal tank are 3 m 2, 12 m 2 and 16 m 2. the capacity of the tank, in litres, is:
24000
Understanding how to calculate the capacity of a cuboidal tank from the areas of its adjacent faces is a common geometry problem. The capacity of a tank is essentially its volume, which for a cuboid is found by multiplying its length, width, and height.
Finding Cuboid Volume from Adjacent Face Areas
A cuboid has three pairs of identical opposite faces. The areas of three adjacent faces correspond to the products of the dimensions: length (l), width (w), and height (h).
- Area of one face = length $\times$ width (lw)
- Area of another adjacent face = width $\times$ height (wh)
- Area of the third adjacent face = height $\times$ length (hl)
The problem gives us the areas of three adjacent faces of the cuboidal tank:
- Area 1: 3 m2
- Area 2: 12 m2
- Area 3: 16 m2
Let's assign these values to the products of the dimensions:
Equation 1: $\text{lw} = 3 \text{ m}^2$
Equation 2: $\text{wh} = 12 \text{ m}^2$
Equation 3: $\text{hl} = 16 \text{ m}^2$
Calculating the Volume of the Cuboidal Tank
The volume (V) of a cuboid is given by the formula:
$\text{V} = \text{lwh}$
We can find the volume by multiplying the three area equations:
$(\text{lw}) \times (\text{wh}) \times (\text{hl}) = 3 \times 12 \times 16$
$\text{l} \times \text{w} \times \text{w} \times \text{h} \times \text{h} \times \text{l} = 576$
$\text{l}^2 \text{w}^2 \text{h}^2 = 576$
We can rewrite the left side as $(\text{lwh})^2$:
$(\text{lwh})^2 = 576$
Since $\text{V} = \text{lwh}$, we have:
$\text{V}^2 = 576$
To find the volume V, we take the square root of 576:
$\text{V} = \sqrt{576}$
$\text{V} = 24$
The volume of the cuboidal tank is 24 cubic meters (m3).
Converting Volume from Cubic Meters to Litres
The question asks for the capacity in litres. We need to convert the volume from cubic meters to litres. The conversion factor is:
1 m3 = 1000 litres
To convert 24 m3 to litres, we multiply by 1000:
Capacity in litres = Volume in m3 $\times$ 1000
Capacity in litres = $24 \times 1000$
Capacity in litres = $24000$
The capacity of the tank is 24000 litres.
| Given Information | Value |
|---|---|
| Area of 1st adjacent face (lw) | 3 m2 |
| Area of 2nd adjacent face (wh) | 12 m2 |
| Area of 3rd adjacent face (hl) | 16 m2 |
| Calculation Step | Formula/Method | Result |
|---|---|---|
| Multiply adjacent areas | $(\text{lw}) \times (\text{wh}) \times (\text{hl}) = \text{l}^2\text{w}^2\text{h}^2$ | $3 \times 12 \times 16 = 576 \text{ m}^6$ |
| Volume squared | $\text{V}^2 = (\text{lwh})^2 = \text{l}^2\text{w}^2\text{h}^2$ | $\text{V}^2 = 576$ |
| Volume in m3 | $\text{V} = \sqrt{\text{V}^2}$ | $\text{V} = \sqrt{576} = 24 \text{ m}^3$ |
| Volume in litres | Volume (m3) $\times$ 1000 | $24 \times 1000 = 24000$ litres |
Conclusion on Tank Capacity
Based on the areas of the three adjacent faces, the calculated volume of the cuboidal tank is 24 m3. Converting this volume to litres gives us the capacity of the tank, which is 24000 litres.
Revision Table: Cuboid Geometry
| Concept | Formula/Definition |
|---|---|
| Cuboid Volume (V) | $\text{V} = \text{length} \times \text{width} \times \text{height}$ |
| Area of a face (e.g., top/bottom) | $\text{Area} = \text{length} \times \text{width}$ |
| Area of a face (e.g., front/back) | $\text{Area} = \text{length} \times \text{height}$ |
| Area of a face (e.g., side) | $\text{Area} = \text{width} \times \text{height}$ |
| Relationship between adjacent areas and volume | If adjacent areas are A1, A2, A3, then $\text{V}^2 = \text{A}_1 \times \text{A}_2 \times \text{A}_3$ |
| Conversion m3 to Litres | 1 m3 = 1000 litres |
Additional Information on Volume and Capacity
Volume: Volume is the amount of three-dimensional space occupied by an object or a region of space. For a cuboid, it measures how much 'space' is inside the tank.
Capacity: Capacity refers to the amount that a container can hold. While closely related to volume, capacity is often used when talking about liquids or gases and is typically measured in units like litres or gallons.
In the metric system, volume and capacity are directly related. The unit m3 is a standard unit of volume, and the litre is a common unit of capacity. The equivalence 1 m3 = 1000 litres is a fundamental conversion often used in practical problems involving tanks, reservoirs, and fluid measurements.
This problem demonstrates a useful trick: if you know the areas of the three faces that meet at a corner of a cuboid, you can easily find the volume by multiplying the areas and taking the square root.
Important Questions from Mensuration
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