If the area of the base of a cuboid is 80 square cm, the area of the other two adjacent faces are 72 square cm and 40 square cm, then the volume of that cuboid is :

Question

Accepted Answer

To determine the volume of the cuboid, we need to use the information given about the areas of its faces. The cuboid has three pairs of opposite faces, and the areas of these faces are given as follows:

1. The area of the base of the cuboid is 80 square cm.
2. The area of one of the adjacent faces is 72 square cm.
3. The area of the other adjacent face is 40 square cm.

Let's denote the dimensions of the cuboid as $ l $, $ w $, and $ h $, where:
- $ l $ and $ w $ are the dimensions of the base, so $ l \times w = 80 $.
- $ l $ and $ h $ are the dimensions of one of the adjacent faces, so $ l \times h = 72 $.
- $ w $ and $ h $ are the dimensions of the other adjacent face, so $ w \times h = 40 $.

To find the volume of the cuboid, we need to calculate $ l \times w \times h $.

From the equations:
1. $ l \times w = 80 $
2. $ l \times h = 72 $
3. $ w \times h = 40 $

We can find the volume by multiplying these three equations together:
$$ (l \times w) \times (l \times h) \times (w \times h) = 80 \times 72 \times 40 $$

This simplifies to:
$$ (l \times w \times h)^2 = 80 \times 72 \times 40 $$

Taking the square root of both sides gives:
$$ l \times w \times h = \sqrt{80 \times 72 \times 40} $$

Calculating the product:
$$ 80 \times 72 \times 40 = 230400 $$

Taking the square root:
$$ \sqrt{230400} = 480 $$

Thus, the volume of the cuboid is 480 cm³, which corresponds to Option 1.

Explanation for why other options are incorrect:
- **Option 2 (360 cm³):** This value does not satisfy the equation derived from the areas of the faces. The calculations clearly show that the volume must be 480 cm³.
- **Option 3 (240 cm³):** Similarly, this value is incorrect as it does not match the calculated volume from the given face areas.
- **Option 4 (80 cm³):** This is far too small to be the volume of the cuboid given the dimensions implied by the face areas.

Therefore, Option 1 is the correct answer, as it is the only option that matches the calculated volume of the cuboid based on the given face areas.