Calculating Percentage Change in Cuboid Volume

This problem involves understanding how percentage changes in the dimensions of a cuboid affect its volume, and then determining the necessary change in one dimension to maintain the original volume.

Let's denote the original dimensions of the cuboid as follows:

• Original Length = \(L\)
• Original Breadth = \(B\)
• Original Height = \(H\)

The original volume of the cuboid is given by the formula:

\[V_{\text{original}} = L \times B \times H\]

Impact of Increasing Length and Breadth

The problem states that both the length and breadth are increased by 12 percent. Let's calculate the new dimensions:

• New Length \(L_{\text{new}}\) = \(L + 12\% \text{ of } L = L + \frac{12}{100}L = L(1 + 0.12) = 1.12L\)
• New Breadth \(B_{\text{new}}\) = \(B + 12\% \text{ of } B = B + \frac{12}{100}B = B(1 + 0.12) = 1.12B\)

Let the new height be \(H_{\text{new}}\). The new volume of the cuboid will be:

\[V_{\text{new}} = L_{\text{new}} \times B_{\text{new}} \times H_{\text{new}} = (1.12L) \times (1.12B) \times H_{\text{new}} = (1.12)^2 \times L \times B \times H_{\text{new}}\]

Calculating \((1.12)^2\):

\[(1.12)^2 = 1.12 \times 1.12 = 1.2544\]

So, the new volume is:

\[V_{\text{new}} = 1.2544 \times L \times B \times H_{\text{new}}\]

Maintaining Constant Volume

The problem requires the volume of the cuboid to remain the same. This means the new volume must be equal to the original volume:

\[V_{\text{new}} = V_{\text{original}}\]

\[1.2544 \times L \times B \times H_{\text{new}} = L \times B \times H\]

We can cancel out \(L \times B\) from both sides (assuming \(L \neq 0\) and \(B \neq 0\), which must be true for a cuboid):

\[1.2544 \times H_{\text{new}} = H\]

Now, we can find the new height \(H_{\text{new}}\) in terms of the original height \(H\):

\[H_{\text{new}} = \frac{H}{1.2544}\]

Calculating Percentage Reduction in Height

To find the percentage reduction in height, we first calculate the decrease in height, which is the original height minus the new height:

\[\text{Decrease in Height} = H - H_{\text{new}} = H - \frac{H}{1.2544}\]

To express this as a percentage reduction relative to the original height, we use the formula:

\[\text{Percentage Reduction} = \frac{\text{Decrease in Height}}{\text{Original Height}} \times 100\%\]

\[\text{Percentage Reduction} = \frac{H - \frac{H}{1.2544}}{H} \times 100\%\]

We can factor out \(H\) from the numerator:

\[\text{Percentage Reduction} = \frac{H \left(1 - \frac{1}{1.2544}\right)}{H} \times 100\%\]

Cancel out \(H\) from the numerator and denominator:

\[\text{Percentage Reduction} = \left(1 - \frac{1}{1.2544}\right) \times 100\%\]

Now, we calculate the value of \(\frac{1}{1.2544}\):

\[\frac{1}{1.2544} \approx 0.797194\]

Substitute this value back into the formula:

\[\text{Percentage Reduction} = (1 - 0.797194) \times 100\%\]

\[\text{Percentage Reduction} \approx 0.202806 \times 100\%\]

\[\text{Percentage Reduction} \approx 20.28\%\]

So, the height should be reduced by approximately 20.28 percent to keep the volume of the cuboid the same after increasing both length and breadth by 12 percent.

Summary of Calculation Steps:

1. Define original dimensions and volume.
2. Calculate new length and breadth after a 12% increase.
3. Set up the equation for the new volume, introducing the new height.
4. Equate the new volume to the original volume to find the relationship between the new height and the original height.
5. Calculate the percentage decrease in height using the formula based on the original and new heights.

Based on our calculation, the required percentage reduction in height is approximately 20.28 percent.

Since \(V_{\text{new}} = V_{\text{original}}\), we have \(1.2544LBH_{\text{new}} = LBH\), which simplifies to \(1.2544 H_{\text{new}} = H\). Therefore, \(H_{\text{new}} = \frac{H}{1.2544}\).

Percentage reduction in height = \(\frac{H - H_{\text{new}}}{H} \times 100\% = \frac{H - \frac{H}{1.2544}}{H} \times 100\% = \left(1 - \frac{1}{1.2544}\right) \times 100\%\).

\[\left(1 - \frac{1}{1.2544}\right) \times 100\% \approx (1 - 0.797194) \times 100\% \approx 0.202806 \times 100\% \approx 20.28\%\]

Revision Table: Cuboid Volume Changes

Understanding how dimensions affect volume is crucial. A percentage increase in dimensions leads to a more significant percentage increase in area or volume.

In our problem, length and breadth are scaled by \(k_l = 1.12\) and \(k_w = 1.12\). We want the overall volume scaling factor \(k_l k_w k_h\) to be 1 (to keep the volume same). So, \(1.12 \times 1.12 \times k_h = 1\), which means \(1.2544 \times k_h = 1\). Thus, \(k_h = \frac{1}{1.2544} \approx 0.7972\). This means the new height is about 0.7972 times the original height, representing a reduction.

Additional Information: Percentage Change Calculations

When a quantity changes from an original value \(O\) to a new value \(N\), the percentage change is calculated as:

\[\text{Percentage Change} = \frac{N - O}{O} \times 100\%\]

If \(N > O\), the percentage change is positive (a percentage increase).

If \(N < O\), the percentage change is negative (a percentage decrease). The percentage decrease is often expressed as a positive value by using the formula:

\[\text{Percentage Decrease} = \frac{O - N}{O} \times 100\%\]

In our case, the original height is \(H\) and the new height is \(H_{\text{new}} = \frac{H}{1.2544}\). Since \(1.2544 > 1\), \(H_{\text{new}} < H\), confirming it's a decrease. The percentage decrease is \(\frac{H - H_{\text{new}}}{H} \times 100\%\).