Question

Stacy wants to build a patio with a small, circular pond in her backyard.The pond will have a 6-foot radius. She also wants to install tiles in the remaining area of the patio. The length of the patio is 13 feet longer than the width.
If the cost of installing tiles is $1 per square foot, and the cost of installing the pond is $0.62 per square foot, then which of the following inequalities can be used to solve for the width, x, of the patio, if Stacy can spend no more than $536 on this project?

Answer 1

First we are going to find the Area of the pond using the area of a circle formula: $A= \pi r^2$
where
$A$ is the area of the circle 
$r$ is the radius of the circle 
We know for our problem that the pond will have a 6-foot radius, so $r=6$. Lets replace that value on our area formula:
$A= \pi r^2$
$A= \pi (6)^2$
$A=113.1ft^{2}$
We know that the cost of installing the pond is $0.62 per square foot, so lets multiply the area we just found by the cost:
Total cost of pund=$(0.62)(113.1)=70.12$ dollars

Now, let $x$ be the width the rectangle, so its length will be $x+13$. Remember that the area of a rectangle is width times length, so:
$A=x(x+13)$
$A=(x^2+13x)ft^2$
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:
$70.12+x^2+13x \leq 536$
$x^2+13x \leq 465.88$

We can conclude that the inequality that can be used to fin the width, $x$, of the patio is $x^2+13x \leq 465.88$