A gardener has 100' of edging. She wants to use it to enclose a 130 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 130, what is the width?
To find the possible lengths that a side of the rectangle can have, we need to consider the perimeter of the rectangle, which is equal to the total amount of edging available.
Let's assume the length of the rectangle is x. The width can be calculated by dividing the area (130 square feet) by the length:
Width = Area / Length
Width = 130 / x
Since the perimeter of a rectangle is equal to the sum of all its sides, we can calculate it using the formula:
Perimeter = 2 * (Length + Width)
We know that the total amount of edging available is 100 feet, so we can set up the equation:
100 = 2 * (Length + Width)
Substituting the expressions for the length and width into the equation, we get:
100 = 2 * (x + 130 / x)
Simplifying further:
100 = 2x + 260 / x
To remove the fraction, we can multiply both sides of the equation by x:
100x = 2x^2 + 260
Rearranging the equation to standard quadratic form:
2x^2 - 100x + 260 = 0
Now, we can solve this quadratic equation to find the possible values for x.
Using the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / (2a)
Plugging in the values:
a = 2, b = -100, c = 260
x = (-(-100) ± sqrt((-100)^2 - 4 * 2 * 260)) / (2 * 2)
x = (100 ± sqrt(10000 - 2080)) / 4
x = (100 ± sqrt(7920)) / 4
Now, we simplify and calculate the possible values for x:
x = (100 ± sqrt(7920)) / 4
x = (100 ± 88.98) / 4
x ≈ (100 + 88.98) / 4 ≈ 197.98 / 4 ≈ 49.5
x ≈ (100 - 88.98) / 4 ≈ 11.02 / 4 ≈ 2.755
However, we need to consider that the length of the rectangle cannot be negative. Therefore, the only valid length is:
x ≈ 2.755
Thus, the possible lengths for a side of the rectangle, using interval notation, is approximately (2.755, 2.755).