A gardener has 72' of edging. She wants to use it to enclose a 125 square foot rectangular area; she does not have to use up all of the edging. What are the possible lengths that a side of the rectangle can have? Answer using interval notation. Hint: If the length of the rectangle is x, and the area is 125, what is the width?
To find the possible lengths of a side of the rectangle, we can use the given information about the edging and the area.
Let's assume the length of the rectangle is x. The width can be calculated by dividing the area by the length:
Width = Area / Length = 125 / x
The perimeter of the rectangle can be represented as:
Perimeter = 2*(Length + Width) = 2*(x + 125/x)
According to the problem, the gardener has 72 feet of edging which means the perimeter should equal 72:
2*(x + 125/x) = 72
Now, we can simplify the equation:
x + 125/x = 36
To solve this equation, we'll multiply through by x to get rid of the fraction:
x^2 + 125 = 36x
Then, rearrange the equation to be written in quadratic form:
x^2 - 36x + 125 = 0
Now, we can factorize or use the quadratic formula to solve for x.
By factoring, we find:
(x - 25)(x - 5) = 0
This implies that either x - 25 = 0 or x - 5 = 0.
From these equations, we can determine the possible values for x:
x = 25 or x = 5
Since we are looking for possible lengths, the answer is given by the interval notation:
[5, 25]