A GARDENER HAS 60' of edging. She wants to use it to enclose a 120 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 120, what is the width?

Question

A GARDENER HAS 60' of edging. She wants to use it to enclose a 120 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 120, what is the width?

Answer 1

x+y <= 60

xy = 120

so,
x + 120/x <= 60
x^2 + 120 <= 60x
x^2 - 60x + 120 <= 0
The two roots of this equation are
x = [60±√(3600-480)]/2 = 30±2√195

since this is a parabola which open up, it is negative between the roots. so,

x ∊ [30-2√195,30+2√195]

Answer 2

To find the possible lengths of a side of the rectangle, we need to use the given information and apply it to the formula for calculating the perimeter of a rectangle.

Let's start by understanding the relationship between the length, width, and area of a rectangle. The formula for finding the area of a rectangle is:

Area = length * width

In this case, the area is given as 120 square feet. So we have:

120 = length * width

Next, we need to express the perimeter of the rectangle using the given information. The formula for finding the perimeter of a rectangle is:

Perimeter = 2 * length + 2 * width

The gardener has 60 feet of edging available, which can be used to enclose the rectangle. So we have:

60 = 2 * length + 2 * width

We can rewrite this equation to solve for width in terms of the length:

60 = 2 * length + 2 * (120 / length)
60 = 2 * length + 240 / length
30 = length + 120 / length

To simplify this equation, we can multiply both sides by the length:

30 * length = length * length + 120
30 * length = length^2 + 120

Rearranging the terms, we have a quadratic equation:

length^2 - 30 * length + 120 = 0

Now we can solve this quadratic equation to find the possible lengths of a side of the rectangle.

Using the quadratic formula, we have:

length = (-b ± √(b^2 - 4ac)) / (2a)

For our equation, a = 1, b = -30, and c = 120. Substituting these values into the formula, we get:

length = (-(-30) ± √((-30)^2 - 4 * 1 * 120)) / (2 * 1)

Simplifying further, we have:

length = (30 ± √(900 - 480)) / 2
length = (30 ± √420) / 2
length = (30 ± √(4 * 105)) / 2
length = (30 ± 2√105) / 2
length = 15 ± √105

Therefore, the possible lengths of a side of the rectangle are 15 + √105 and 15 - √105. We can express this in interval notation as:

[15 - √105, 15 + √105]