Two concentric circles have radii x and y , where y

Two concentric circles have radii $ x $ and $ y $, where $ y > x $. The area between the circles must be at least 10 square units. (a) Find a system of inequalities describing the constraints on the circles. (b) Use a graphing utility to graph the system of inequalities in part (a). Graph the line $ y = x $ in the same viewing window. (c) Identify the graph of the line in relation to the boundary of the inequality. Explain its meaning in the context of the problem.

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

(a) Find a system of inequalities describing the constraints on the circles.
(b) Use a graphing utility to graph the system of inequalities in part (a). Graph the line $ y = x $ in the same viewing window.
(c) Identify the graph of the line in relation to the boundary of the inequality. Explain its meaning in the context of the problem.

Key Concepts

Concentric Circles

Concentric circles are circles that share the same center point but have different radii. In mathematical problems, they are significant because comparing their properties (such as the difference of their areas) can lead to studies of annulus areas and other related geometric regions.

Area of an Annulus

The area between two concentric circles, known as an annulus, is computed by subtracting the area of the smaller circle from the area of the larger circle, yielding ?(y^2 - x^2) when the circles have radii x and y respectively. This concept is key when establishing constraints based on the difference in areas in geometric problems.

Inequalities Involving Areas

When a problem specifies a minimum area between shapes such as concentric circles, it results in forming an inequality. For example, setting ?(y^2 - x^2) ? A (where A is the area threshold) is a common method to express such a geometric constraint mathematically, which is essential for defining the solution set.

Systems of Inequalities

A system of inequalities is a collection of two or more inequalities that are considered together. In this context, the system may include the inequality regarding the area of the annulus as well as additional inequalities like y > x, ensuring that the larger circle’s radius is actually greater than the smaller one.

Graphical Interpretation of Inequalities

Graphing inequalities involves plotting the regions corresponding to each inequality on a coordinate plane and identifying their intersections. Plotting the line y = x, which represents the boundary between x and y, is useful for visually confirming that the selected points obey the condition y > x, thereby separating acceptable and unacceptable regions in the context of the problem.

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