Cuboid Volume Calculation Explained
This problem asks us to find the overall percentage increase in the volume of a cuboid after its dimensions (length, breadth, and height) are increased by different percentages.
Understanding Cuboid Volume
The volume of a cuboid is calculated by multiplying its length, breadth, and height.
If the original dimensions are L (length), B (breadth), and H (height), the original volume (\(V_1\)) is given by:
\( V_1 = L \times B \times H \)
Calculating New Dimensions
The problem states the following increases:
- Length increases by 10%.
- Breadth increases by 20%.
- Height increases by 50%.
Let's calculate the new dimensions (\(L'\), \(B'\), \(H'\)):
- New Length (\(L'\)): The original length L is increased by 10%.
\( L' = L + (10\% \text{ of } L) = L + \frac{10}{100} L = L(1 + 0.10) = 1.1 L \) - New Breadth (\(B'\)): The original breadth B is increased by 20%.
\( B' = B + (20\% \text{ of } B) = B + \frac{20}{100} B = B(1 + 0.20) = 1.2 B \) - New Height (\(H'\)): The original height H is increased by 50%.
\( H' = H + (50\% \text{ of } H) = H + \frac{50}{100} H = H(1 + 0.50) = 1.5 H \)
Calculating New Volume
The new volume (\(V_2\)) is calculated using the new dimensions:
\( V_2 = L' \times B' \times H' \)
Substituting the expressions for \(L'\), \(B'\), and \(H'\):
\( V_2 = (1.1 L) \times (1.2 B) \times (1.5 H) \)
Rearranging the terms:
\( V_2 = (1.1 \times 1.2 \times 1.5) \times (L \times B \times H) \)
First, calculate the product of the multipliers:
\( 1.1 \times 1.2 = 1.32 \)
\( 1.32 \times 1.5 = 1.98 \)
So, the new volume is:
\( V_2 = 1.98 \times (L \times B \times H) \)
Since \(V_1 = L \times B \times H\), we have:
\( V_2 = 1.98 V_1 \)
Determining Percentage Increase
The increase in volume is \(V_2 - V_1\).
The percentage increase is calculated as:
\( \text{Percentage Increase} = \frac{V_2 - V_1}{V_1} \times 100\% \)
Substitute \(V_2 = 1.98 V_1\):
\( \text{Percentage Increase} = \frac{1.98 V_1 - V_1}{V_1} \times 100\% \)
\( \text{Percentage Increase} = \frac{(1.98 - 1) V_1}{V_1} \times 100\% \)
\( \text{Percentage Increase} = \frac{0.98 V_1}{V_1} \times 100\% \)
\( \text{Percentage Increase} = 0.98 \times 100\% \)
\( \text{Percentage Increase} = 98\% \)
Conclusion
The volume of the cuboid increases by 98%.
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