The surface areas of the six faces of a rectangular solid are 16, 16, 32, 32, 72 and 72 square centimetres. The volume of the solid, in cubic centimetres, is

Options

A. 192
B. 384
C. 480
D. 2592

Answer

To find the volume of a rectangular solid (cuboid) given the surface areas of its six faces, we can follow these steps:

### Step 1: Identify the Surface Areas
The surface areas of the six faces are given as:
- 16 cm² (two faces)
- 32 cm² (two faces)
- 72 cm² (two faces)

### Step 2: Assign Variables
Let:
- Length = \( l \)
- Breadth = \( b \)
- Height = \( h \)

From the surface areas, we can set up the following equations based on the areas of the faces:
1. \( lb = 16 \) (for the top and bottom faces)
2. \( lh = 32 \) (for the front and back faces)
3. \( bh = 72 \) (for the left and right faces)

### Step 3: Solve the Equations
We have three equations:
1. \( lb = 16 \) (Equation 1)
2. \( lh = 32 \) (Equation 2)
3. \( bh = 72 \) (Equation 3)

#### Step 3.1: Express \( h \) in terms of \( b \)
From Equation 1:
\[ h = \frac{32}{l} \]
Substituting \( h \) in Equation 3:
\[ b \left(\frac{32}{l}\right) = 72 \]
\[ \frac{32b}{l} = 72 \]
\[ b = \frac{72l}{32} = \frac{9l}{4} \]

#### Step 3.2: Substitute \( b \) back into Equation 1
Substituting \( b \) in Equation 1:
\[ l \left(\frac{9l}{4}\right) = 16 \]
\[ \frac{9l^2}{4} = 16 \]
\[ 9l^2 = 64 \]
\[ l^2 = \frac{64}{9} \]
\[ l = \frac{8}{3} \text{ cm} \]

#### Step 3.3: Find \( b \)
Now substituting \( l \) back into the equation for \( b \):
\[ b = \frac{9l}{4} = \frac{9 \times \frac{8}{3}}{4} = \frac{72}{12} = 6 \text{ cm} \]

#### Step 3.4: Find \( h \)
Now substituting \( l \) back into Equation 2 to find \( h \):
\[ h = \frac{32}{l} = \frac{32}{\frac{8}{3}} = 32 \times \frac{3}{8} = 12 \text{ cm} \]

### Step 4: Calculate the Volume
The volume \( V \) of the cuboid is given by:
\[ V = l \times b \times h \]
Substituting the values we found:
\[ V = \left(\frac{8}{3}\right) \times 6 \times 12 \]
\[ V = \frac{8 \times 6 \times 12}{3} = \frac{576}{3} = 192 \text{ cm}^3 \]

### Final Answer
The volume of the rectangular solid is **192 cubic centimeters**.

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