First we are going to find the Area of the pond using the area of a circle formula: [tex]A= \pi r^2[/tex]
where
[tex]A[/tex] is the area of the circle
[tex]r[/tex] is the radius of the circle
We know for our problem that the pond will have a 6-foot radius, so [tex]r=6[/tex]. Lets replace that value on our area formula:
[tex]A= \pi r^2[/tex]
[tex]A= \pi (6)^2[/tex]
[tex]A=113.1ft^{2}[/tex]
We know that the cost of installing the pond is $0.62 per square foot, so lets multiply the area we just found by the cost:
Total cost of pund=[tex](0.62)(113.1)=70.12[/tex] dollars
Now, let [tex]x[/tex] be the width the rectangle, so its length will be [tex]x+13[/tex]. Remember that the area of a rectangle is width times length, so:
[tex]A=x(x+13)[/tex]
[tex]A=(x^2+13x)ft^2[/tex]
Since the cost of installing ties is $1, the cost of installing ties in our rectangle will be x^2+13x dollars.
Stacy can spend no more than $536 on this project, so we can setup an inequality relating the cost of the pound and the cost of installing ties:
[tex]70.12+x^2+13x \leq 536[/tex]
[tex]x^2+13x \leq 465.88[/tex]
We can conclude that the inequality that can be used to fin the width, [tex]x[/tex], of the patio is [tex]x^2+13x \leq 465.88[/tex]
