The percentage increase in the area of a square when all its sides are elongated by 10% is 21%.
Understanding Area Scaling
When the sides of a square are increased by a certain percentage, its area expands by a greater percentage. This is because the area is a two-dimensional measurement, calculated by multiplying the side length by itself. Therefore, any change in the side length is compounded when determining the area.
Let's break down the calculation to demonstrate this principle:
Step-by-Step Calculation
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Define Original Dimensions:
- Let 's' represent the length of each side of the original square.
- The original area (A_original) of the square is found by the formula:
A_original = s * s = s².
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Calculate New Dimensions After Elongation:
- The problem states that all sides are elongated by 10%. This means the new side length (s_new) will be the original side length plus 10% of the original side length.
s_new = s + (0.10 * s) = 1.10s.
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Determine the New Area:
- The new area (A_new) of the larger square is calculated using the new side length:
A_new = (1.10s) * (1.10s) = (1.10)²s² = 1.21s².
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Calculate the Increase in Area:
- The actual increase in area (A_increase) is the difference between the new area and the original area:
A_increase = A_new - A_original = 1.21s² - s² = 0.21s².
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Compute the Percentage Increase:
- To find the percentage increase, divide the increase in area by the original area and multiply by 100:
Percentage Increase = (A_increase / A_original) * 100Percentage Increase = (0.21s² / s²) * 100Percentage Increase = 0.21 * 100 = 21%
This calculation reveals that a 10% linear increase in the side of a square results in a 21% increase in its overall area. This result is consistent irrespective of the initial size of the square.
Illustrative Example
To provide a clearer understanding, consider a square with an initial side length:
| Property | Original Square | New Square (After 10% Elongation) |
|---|---|---|
| Side Length (s) | 10 units | 11 units (10 + 10 * 0.10) |
| Area (s²) | 100 sq units | 121 sq units (11 * 11) |
| Increase in Area | - | 21 sq units (121 - 100) |
| Percentage Increase | - | (21 / 100) * 100 = 21% |
Practical Relevance
Understanding how changes in linear dimensions affect area is vital in numerous real-world applications:
- Construction and Real Estate: When scaling up building plans or calculating floor space, even minor increases in dimensions can lead to significantly larger areas, impacting material costs (e.g., flooring, roofing) and property values.
- Manufacturing and Engineering: In designing products, engineers must account for how changes in dimensions affect surface area, which can influence factors like heat dissipation, material usage, and weight.
- Graphics and Design: Resizing images or designs requires an understanding of how scaling affects the overall area and aspect ratios to maintain visual integrity and quality.
- Personal Finance: When considering purchases like larger carpets or garden plots, knowing the area's exponential growth relative to linear dimensions can help in budgeting and planning.
The general rule is that if a linear dimension of a two-dimensional shape is increased by a factor of k, its area increases by a factor of k². In this case, a 10% increase means k = 1.10. Therefore, the area increases by (1.10)² = 1.21, which represents a 21% increase from the original area.
