Question

the faces of a cuboid have the areas shown below. the edges of the cuboid all have integer lengths.

what is the volume of the cuboid?

not drawn accurately

Explanation:

⚡ Using what you learned: calculating volume of rectangular prisms

Step 1: Set up equations for the face areas

Let the integer edge lengths of the cuboid be \( a \), \( b \), and \( c \). The areas of the three adjacent faces are given by:

$$ a \times b = 28 $$

$$ b \times c = 12 $$

$$ a \times c = 21 $$

Step 2: Find the integer edge lengths

Since \( a \), \( b \), and \( c \) must be integers, we find their values by looking at the common factors:

• \( b \) is a common factor of \( 28 \) and \( 12 \), so \( b = 4 \).
• \( a \) is a common factor of \( 28 \) and \( 21 \), so \( a = 7 \).
• \( c \) is a common factor of \( 12 \) and \( 21 \), so \( c = 3 \).

Checking the products:

• \( a \times b = 7 \times 4 = 28 \)
• \( b \times c = 4 \times 3 = 12 \)
• \( a \times c = 7 \times 3 = 21 \)

The dimensions are \( 7\text{ m} \), \( 4\text{ m} \), and \( 3\text{ m} \).

Step 3: Calculate the volume

The volume \( V \) of the cuboid is:

$$ V = a \times b \times c $$

$$ V = 7 \times 4 \times 3 = 84\text{ m}^3 $$

Alternatively, using the relationship \( V^2 = (ab) \times (bc) \times (ac) \):

$$ V^2 = 28 \times 12 \times 21 = 7056 $$

$$ V = \sqrt{7056} = 84\text{ m}^3 $$

Answer:

\( 84\text{ m}^3 \)