Exploring the Area Increase of a Rectangle After Length and Breadth are Extended
In this article, we will delve into the process of determining how the area of a rectangle changes when both its length and breadth are extended. We'll examine the calculations for specific values as well as provide a general formula to solve this geometric problem. Understanding these concepts can be particularly useful for students, mathematicians, and anyone involved in designing or measuring physical spaces.
Problem Statement
Consider a square with side length s. When the side length and breadth of this square are extended by 30% and 20% respectively, how much does the new area surpass the original area?
Step-by-Step Calculations
Let's denote the side length of the original square as s.
The area of the square is given by:
Area_{square} s^2
After the extensions, the new dimensions can be expressed as:
New Length 1.30s
New Breadth 1.20s
The area of the resulting rectangle can now be calculated as:
Area_{rectangle} 1.30s times 1.20s 1.56s^2
Now, we can find the difference in the area:
Area_{difference} Area_{rectangle} - Area_{square} 1.56s^2 - s^2 0.56s^2
Therefore, the area of the new rectangle exceeds the original square's area by
0.56s^2.
Example Calculations
Let's apply this calculation for a square with an initial side length of 10 units.
Original Area 10 u00d7 10 100 square units.
After increasing the length to 14 units and the breadth to 13 units:
Area 14 u00d7 13 182 square units.
The increase in area is:
182 - 100 82 square units.
Notice that this increase applies to any starting size. For instance, if the original dimensions were 20 units each, the new dimensions would be 26 units and 24 units, and the calculation would yield the same difference.
General Formula Consideration
Let's denote the length and breadth of the original square as
l and b.
After extending the sides, the new dimensions will be:
Length 1.40s
Breadth 1.20s
The new area of the rectangle can be calculated as:
Area_{new} 1.40l times 1.20b 1.68lb
The increase in area from the original square can then be expressed as:
Area_{increase} 1.68lb - lb 0.68lb
Therefore, the area surpasses the original area by
68% of LB.
Conclusion and Practical Applications
Understanding how to determine the increase in area after extending the dimensions of a rectangle is valuable in various fields, including architecture, engineering, and design. These calculations can help in assessing the impact of changes in size on the overall structure or space. Whether dealing with small or large dimensions, the principles remain consistent, aiding in efficient problem-solving and decision-making.