The area of a rectangular plot of land is [tex]$\left(x^2+13 x+40\right)$[/tex] sq. m.
(i) Find the length and breadth of the land.
(ii) If the length and breadth of the land are reduced by [tex]$2 / 2 m$[/tex] respectively, find the new area of the land.
Answer (1)
Factor the area expression x 2 + 13 x + 40 to find the length and breadth: ( x + 5 ) ( x + 8 ) .
Reduce the length and breadth by 2 meters: new length is ( x + 3 ) and new breadth is ( x + 6 ) .
Calculate the new area by multiplying the new length and breadth: ( x + 3 ) ( x + 6 ) = x 2 + 9 x + 18 .
The new area of the land is x 2 + 9 x + 18 sq. m.
Explanation
Understanding the Problem We are given the area of a rectangular plot of land as x 2 + 13 x + 40 square meters. Our goal is to find the length and breadth of the land and then determine the new area if both the length and breadth are reduced by 2 meters.
Factoring the Quadratic Expression To find the length and breadth, we need to factorize the quadratic expression x 2 + 13 x + 40 . We are looking for two numbers that add up to 13 and multiply to 40. These numbers are 5 and 8.
Finding Length and Breadth Therefore, we can write the area as ( x + 5 ) ( x + 8 ) . This means the length and breadth of the land are ( x + 5 ) meters and ( x + 8 ) meters, respectively.
Reducing Length and Breadth Now, we reduce both the length and breadth by 2 meters. The new length is ( x + 5 − 2 ) = ( x + 3 ) meters, and the new breadth is ( x + 8 − 2 ) = ( x + 6 ) meters.
Calculating the New Area The new area of the land is the product of the new length and new breadth, which is ( x + 3 ) ( x + 6 ) . Expanding this expression, we get: ( x + 3 ) ( x + 6 ) = x 2 + 6 x + 3 x + 18 = x 2 + 9 x + 18 So, the new area of the land is ( x 2 + 9 x + 18 ) square meters.
Final Answer Therefore, (i) the length and breadth of the land are ( x + 5 ) meters and ( x + 8 ) meters, and (ii) the new area of the land is ( x 2 + 9 x + 18 ) square meters.
Examples
Understanding area calculations is crucial in various real-life scenarios. For instance, when planning a garden, knowing the area helps determine the amount of soil needed and the number of plants that can fit. Similarly, in construction, calculating the area of a room is essential for estimating the amount of flooring material required. By applying these mathematical concepts, we can efficiently manage resources and make informed decisions in everyday tasks.
