a) The area of a rectangular field is [tex]$\left(x^2+8 x+15\right)$[/tex] sq. m.
(i) Find the length and breadth of the field.
(ii) Find the perimeter of the field.
Answer (1)
Factor the quadratic expression x 2 + 8 x + 15 to find the length and breadth: ( x + 3 ) and ( x + 5 ) .
Apply the perimeter formula P = 2 ( l + b ) with l = ( x + 5 ) and b = ( x + 3 ) .
Simplify the expression: P = 2 (( x + 5 ) + ( x + 3 )) = 2 ( 2 x + 8 ) .
The perimeter of the field is 4 x + 16 meters.
Explanation
Problem Analysis We are given the area of a rectangular field as x 2 + 8 x + 15 square meters. Our goal is to find the length and breadth of the field and then determine its perimeter.
Factoring the Area To find the length and breadth, we need to factor the quadratic expression representing the area. Factoring x 2 + 8 x + 15 involves finding two numbers that multiply to 15 and add up to 8. These numbers are 3 and 5. Therefore, we can write the area as ( x + 3 ) ( x + 5 ) .
Identifying Length and Breadth So, the length and breadth of the rectangular field can be represented as ( x + 5 ) meters and ( x + 3 ) meters, respectively.
Applying the Perimeter Formula Now, we need to find the perimeter of the field. The formula for the perimeter of a rectangle is P = 2 ( l + b ) , where l is the length and b is the breadth. Substituting the expressions for length and breadth, we get:
P = 2 (( x + 5 ) + ( x + 3 ))
Simplifying the Perimeter Expression Simplifying the expression, we have:
P = 2 ( x + 5 + x + 3 )
P = 2 ( 2 x + 8 )
P = 4 x + 16 meters.
Final Answer Therefore, the perimeter of the rectangular field is ( 4 x + 16 ) meters.
Examples
Understanding how to calculate the dimensions and perimeter of a rectangular area is useful in many real-world scenarios. For example, if you're planning a garden, knowing the area helps you determine how much soil you need, while the perimeter helps you calculate how much fencing to buy. Similarly, architects and construction workers use these calculations to design and build structures efficiently, ensuring they have the right amount of materials.
