Which of the following represents the area of a rectangle whose length is $x+1$ and whose width is $x+11$?
A. $x^2+10 x+11$
B. $x^2+11 x+12$
C. $x^2+11$
D. $x^2+12 x+11$
Answer (1)
The area of a rectangle is the product of its length and width.
Given length x + 1 and width x + 11 , the area is ( x + 1 ) ( x + 11 ) .
Expanding the expression gives x 2 + 11 x + x + 11 = x 2 + 12 x + 11 .
The area of the rectangle is x 2 + 12 x + 11 .
Explanation
Understanding the Problem We are given a rectangle with length x + 1 and width x + 11 . We need to find the expression that represents the area of this rectangle.
Applying the Area Formula The area of a rectangle is given by the formula:
Area = length × width
In this case, the length is x + 1 and the width is x + 11 . So, we have:
Area = ( x + 1 ) ( x + 11 )
Expanding the Expression To find the expression for the area, we need to expand the product ( x + 1 ) ( x + 11 ) . We can use the distributive property (also known as the FOIL method) to do this:
( x + 1 ) ( x + 11 ) = x ( x + 11 ) + 1 ( x + 11 )
= x 2 + 11 x + x + 11
= x 2 + 12 x + 11
Final Answer Therefore, the area of the rectangle is represented by the expression x 2 + 12 x + 11 .
Examples
Understanding how to calculate the area of a rectangle with variable side lengths is useful in many real-world scenarios. For example, if you're designing a rectangular garden where one side is 1 meter longer than a variable length 'x' and the other side is 11 meters longer than 'x', you can use the expression x 2 + 12 x + 11 to determine the total area of the garden. This helps in planning how much space you have for planting and how much fencing you'll need.
