Chapter 7.5, Problem 76E

(a)

To determine

To write: a system of inequalities that describes the constraints on the circles.

Answer to Problem 76E

  $\begin{array}{c}\pi y^2-\pi x^2\ge 10\\ y>x\\ x>0\\ y>0\end{array}$

Explanation of Solution

Given information:

Two concentric circles have radii $x$ and $y$ , where $y>x$ . The area between the circles is as least 10 square is at least 10 square units.

Calculation:

Let us consider the two circle of radius $x$ and $y$ where $y>x$ .

The area of larger circle is $\pi y^2$ .

The area of smaller circle is $\pi x^2$ .

Since the area between the circles must be at least 10 square units, therefore we have

  $\pi y^2-\pi x^2\ge 10$

Also,

  $y>x$

And redii are positive. Therefore, the system of inequalities is

  $\begin{array}{c}\pi y^2-\pi x^2\ge 10\\ y>x\\ x>0\\ y>0\end{array}$

(b)

To determine

To graph: the system of inequalities in part (a). Graph the line $y=x$ in the same viewing window.

Explanation of Solution

Consider the system from part (a).

  $\begin{array}{c}\pi y^2-\pi x^2\ge 10\\ y>x\\ x>0\\ y>0\end{array}$

The graph of the system is shown below.