Percentage Increase in Volume When Each Edge of a Cuboid is Increased by 20%

When we increase each edge of a cuboid by 20%, we are essentially making each dimension 1.2 times its original value. This transformation can significantly affect the volume of the cuboid, leading to a substantial increase in its volume. This article will explore how to calculate this percentage increase in volume step by step.

Understanding the Original Volume

Let’s begin with the original dimensions of a cuboid. Denote the original length, width, and height as l, w, and h, respectively. The original volume V of this cuboid can be expressed as:

[V l times w times h]

Modified Dimensions

When each edge is increased by 20%, the new dimensions are as follows:

l' l times 1.2 w' w times 1.2 h' h times 1.2

With these new dimensions, the new volume of the cuboid, V', becomes:

[V' l' times w' times h' (l times 1.2) times (w times 1.2) times (h times 1.2)]

Expanding the right side of the equation, we get:

[V' l times w times h times (1.2)^3 V times (1.2)^3]

Calculating the Increase in Volume

The cubed value of 1.2 is:

[1.2^3 1.728]

Thus, the new volume can be expressed as:

[V' 1.728 times V]

To find the increase in volume:

[text{Increase in volume} V' - V V times (1.728 - 1) V times 0.728]

Finding the Percentage Increase

To find the percentage increase, we divide the increase in volume by the original volume and multiply by 100:

[text{Percentage increase} left( frac{V' - V}{V} right) times 100 0.728 times 100 72.8%]

Hence, the increase in the percentage of the volume of the cuboid is 72.8%.

Comparison with Cube Volume Increase

For a cube, the increase in volume when each edge is increased by a small percentage (e.g., 20%) can be explored differently. When each edge of a cube is increased by 20%, the new side length is 1.2 times the original side length. Let the original side be a. The original volume V is V a^3. The new side length a' is 1.2a, and the new volume V' is:

[V' (1.2a)^3 1.728a^3 1.728V]

The percentage increase can be calculated as:

[text{Percentage increase} left( frac{V' - V}{V} right) times 100 (1.728 - 1) times 100 72.8%]

Conclusion

Whether dealing with a cuboid or a cube, if each edge is increased by 20%, the volume percentage increase is consistently 72.8%. This percentage increase is robust and remains the same regardless of the original dimensions.

Additional Insights

In practical applications, such calculations can be useful in various fields, including structural engineering, material science, and volume optimization. Understanding how geometric transformations affect volume can be crucial in design and manufacturing processes.

Related Keywords

volume increase cuboid edge increase