The length of a rectangle is 4 centimeters less than its width. What are the dimensions of the rectangle if its area is 117 square centimeters?
usually the length is greater than the width, but if we take the larger dimension to be x, then we have
x(x-4) = 117
Hint: 117 = 9*13
L = W - 4
W(W - 4) = 117 ... W^2 - 4 W - 117 = 0
factor to find W
To solve this problem, we can set up an equation and use algebra to find the dimensions of the rectangle.
Let's assume that the width of the rectangle is x centimeters. According to the problem, the length is 4 centimeters less than the width. So the length would be (x - 4) centimeters.
To find the area of the rectangle, we use the formula: Area = Length * Width. In this case, the area is given as 117 square centimeters. So we have the equation:
(x - 4) * x = 117
Now we can solve this quadratic equation to find the value of x.
Expanding the equation, we get:
x^2 - 4x = 117
Rearranging the terms, we get:
x^2 - 4x - 117 = 0
Now we can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. In this case, let's use the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / (2a)
For our equation, a = 1, b = -4, and c = -117. Plugging these values into the quadratic formula, we get:
x = (-(-4) ± sqrt((-4)^2 - 4(1)(-117))) / (2(1))
x = (4 ± sqrt(16 + 468)) / 2
x = (4 ± sqrt(484)) / 2
x = (4 ± 22) / 2
Now we have two possible values for x:
1. x = (4 + 22) / 2 = 26 / 2 = 13
2. x = (4 - 22) / 2 = -18 / 2 = -9
Since the width of a rectangle cannot be negative, we reject the second solution. Therefore, the width of the rectangle is 13 centimeters.
Using the given information that the length is 4 centimeters less than the width, we can calculate the length:
Length = Width - 4
Length = 13 - 4
Length = 9 centimeters
So, the dimensions of the rectangle are 13 centimeters by 9 centimeters.