the length of a rectangle is 2cm longer than the width. if its length and width are both increased by 4cm, its area is increased by 72 cm^2. find the dimensions of the original rectangle.
To find the dimensions of the original rectangle, let's represent the width of the rectangle as "x" cm.
According to the given information, the length of the rectangle is 2 cm longer than the width. Therefore, the length can be represented as "x + 2" cm.
The area of a rectangle is given by the formula: Area = Length x Width.
So, the area of the original rectangle is:
Original Area = (x + 2) cm x x cm
= x(x + 2) cm^2
When both the length and width are increased by 4 cm, the new length and width become:
New Length = (x + 2) cm + 4 cm = (x + 6) cm
New Width = x cm + 4 cm = (x + 4) cm
The area of the new rectangle is given as increased by 72 cm^2 compared to the original area.
New Area = Original Area + 72 cm^2
(x + 6)(x + 4) = x(x + 2) + 72
Now, let's solve the equation step-by-step:
Expanding both sides:
(x^2 + 6x + 4x + 24) = (x^2 + 2x) + 72
Simplifying:
x^2 + 10x + 24 = x^2 + 2x + 72
Canceling out x^2 from both sides:
10x + 24 = 2x + 72
Moving variables to one side and constants to the other side:
10x - 2x = 72 - 24
8x = 48
Dividing both sides by 8:
x = 6
Therefore, the original width of the rectangle is 6 cm.
Using this value, we can find the original length:
Length = Width + 2
Length = 6 + 2
Length = 8
So, the dimensions of the original rectangle are:
Width = 6 cm
Length = 8 cm