The Percentage Change in Volume of a Cuboid after Altering Dimensions

When dealing with the volume of geometric shapes, understanding the impact of changes in dimensions can be both an interesting and practical problem. In this article, we will explore the scenario where the length, breadth, and height of a cuboid are altered, and how this affects the volume. We will illustrate the process with specific examples before summarizing the general method for such calculations.

Example 1: Initial Dimensions Increase

Consider a cuboid with initial dimensions of length lb and height h. If the length and breadth are increased by 5% each while the height is increased by 100%, we need to determine the percentage change in the volume.

First, let us denote the initial dimensions as follows: Length l Breadth b Height h Thus, the initial volume is Vinitial lbh. Now, the new length and breadth will be: New Length l 5% of l l * 1.05 New Breadth b 5% of b b * 1.05 The new height will be h 100% of h 2h Effectively, the new volume is given by: New Volume (1.05l) * (1.05b) * (2h) 2.205lbh The change in volume is therefore: 2.205lbh - lbh 1.205lbh The percentage change in volume is: (Change in volume / Original Volume) * 100 (1.205lbh / lbh) * 100 120.5%

Generalizing the Method

To generalize this method, let the original lengths of the cuboid be x, y, and z. The general volume is given by xyz. If the length, breadth, and height are increased by 50%, 120%, and 150%, respectively, the new dimensions are 1.5x, 1.2y, and 1.5z. The new volume is calculated as:

New Volume (1.5x) * (1.2y) * (1.5z) 1.5 * 1.2 * 1.5 * xyz 2.7xyz

The change in volume is therefore 2.7xyz - xyz 1.7xyz, and the percentage change is:

(Change in volume / Original Volume) * 100 (1.7xyz / xyz) * 100 170%

Applying the Method to Practical Examples

If l 10, b 10, and h 10: Initial Volume 1000 New Length 11, New Breadth 12, New Height 15 New Volume 11 * 12 * 15 1980 Percentage Change (1980 - 1000) / 1000 * 100 98% Let l, b, and h be symbolic: Initial Volume lbh New Length 1.1l, New Breadth 1.2b, New Height 1.5h New Volume (1.1l) * (1.2b) * (1.5h) 1.98lbh Change in Volume 1.98lbh - lbh 0.98lbh Percentage Change (0.98lbh / lbh) * 100 98%

Conclusion

In summary, the percentage change in volume when altering dimensions of a cuboid can be calculated using the above methods. The key is to understand the proportional increase in each dimension and how this affects the overall volume. Given the example, it is clear that a straightforward calculation approach simplifies the process of determining the percentage change in volume.