Problem 68 - Calculus Early Transcendentals

Question

\(\mathrm{A}\) lune is a crescent-shaped region bounded by the arcs of two circles. Let \(C_{1}\) be a circle of radius 4 centered at the origin. Let \(C_{2}\) be a circle of radius 3 centered at the point (2,0) Find the area of the lune (shaded in the figure) that lies inside \(C_{1}\) and outside \(C_{2}\)

Step-by-Step Solution

Answer

Answer: The area of the lune is \(\frac{7}{24}(31\pi - 7)\).

Step 1: Find the distance between the centers of the circles

We'll start by finding the distance between the centers of the circles, which will help us determine the angle of the sectors created when the circles intersect. We'll use the distance formula for this step. Given points (0,0) and (2,0), the distance is: \(d = \sqrt{(2 - 0)^2 + (0 - 0)^2}= \sqrt{2^2} = 2\)

Step 2: Find the cosine of the angle between the radii

Next, we'll use the Law of Cosines in order to find the angle between the radii of both circles where they intersect. Using the triangle formed by the centers of the circles O and P and the intersection point of the radii, A, we have: \(OA^2 = OP^2 + AP^2 - 2(OP)(AP)\cos{(\angle OAP)}\) Solve the equation for the angle \(\angle OAP\): \(\cos{(\angle OAP)} = \frac{OP^2 + AP^2 - OA^2}{2(OP)(AP)}\) Substitute the given values: \(\cos{(\angle OAP)} = \frac{3^2 + 4^2 - 2^2}{2(3)(4)} = \frac{25}{24}\)

Step 3: Find the areas of both segments

Now that we know the cosine of the angle between the radii, we can find the areas of both segments. We'll use the formula for the area of the segment of a circle: \(A_{seg} = \frac{1}{2}r^2(\theta - \sin{\theta})\) Where \(r\) is the radius of the circle and \(\theta\) is the angle at the intersection point of the radii. Note that \(\sin{\theta}\) can be found using the Pythagorean identity: \(\sin{\theta}=\sqrt{1-\cos^2{(\theta)}}=\sqrt{1-(\frac{25}{24})^2} = \frac{7}{24}\) Determine the area of the segment for Circle 1: \(A_{seg, C_1} = \frac{1}{2}(4^2)((2\arccos(\frac{25}{24})) - (\frac{7}{24})) = \frac{1}{2}(16)(\frac{31\pi}{12} - \frac{7}{12})\) Determine the area of the segment for Circle 2: \(A_{seg, C_2} = \frac{1}{2}(3^2)(2\arccos(\frac{25}{24}) - \frac{7}{24}) = \frac{1}{2}(9)(\frac{31\pi}{12} - \frac{7}{12})\)

Step 4: Calculate the area of the lune

Finally, subtract the area of the smaller segment (Circle 2) from the area of the larger segment (Circle 1) to get the area of the lune: \(Lune Area = A_{seg, C_1} - A_{seg, C_2}\) \(=\frac{1}{2}(16)(\frac{31\pi}{12} - \frac{7}{12}) - \frac{1}{2}(9)(\frac{31\pi}{12} - \frac{7}{12}) \) \(= \frac{1}{2}(7)(\frac{31\pi}{12} - \frac{7}{12}) \) \(= \frac{7}{24}(31\pi - 7)\) Thus, the area of the lune is \(\frac{7}{24}(31\pi - 7)\).

Key Concepts

Circle Geometry

Circle geometry is all about understanding the properties and relationships between circles. It includes studying elements such as radius, diameter, circumference, and areas, as well as how circles interact with each other, such as in the case of intersections. In our exercise, we're dealing with two circles intersecting, creating a look known as a "lune," which is a crescent-shaped region.

The essential features of a circle include:

• The center point, from which all points on the circle are equidistant.
• The radius, which is the distance from the center to any point on the circle.
• The diameter, which is twice the radius, measuring the widest distance across the circle.
• The circumference, the total distance around the circle, which can be calculated using the formula \(C = 2\pi r\).
• The area, calculated as \(A = \pi r^2\).

When two circles intersect, they form an overlapping area and sometimes produce a lune, as seen in our problem. Calculating the specific areas and interactions requires deep understanding of geometry, especially when it involves calculating segments of the circle using angles, as we see in the exercise. Understanding these aspects helps determine how much of one circle lies within another, and how much is excluded.

Law of Cosines

The Law of Cosines is a valuable rule when dealing with triangles, especially non-right triangles, and it extends the well-known Pythagorean theorem. It helps us find unknown lengths and angles when certain other measurements are known. In our exercise, the Law of Cosines is crucial for determining the angles between intersecting circles.

For any triangle with sides labeled as \(a\), \(b\), and \(c\), and the angle opposite side \(c\) being \(\gamma\), the Law of Cosines states:\[ c^2 = a^2 + b^2 - 2ab\cos(\gamma) \]This equation helps solve for an obtuse or acute angle in a triangle when the length of all three sides is known. In the problem, we applied it to the triangle formed by the centers of the circles and their intersection point to discover the angle that plays into quantifying the segments in each circle.

The Law of Cosines thus bridges our understanding from just dealing with circle geometry to integrating concepts of trigonometry, specifically in animalizing the angular relationships within intersecting circles, enabling us to make precise calculations about areas of interest like that of a lune.

Intersection of Circles

The intersection of two circles refers to the points or areas where the circles overlap. Understanding this intersection is vital in geometry problems involving space and area calculations, as it often results in more complex shapes like lunes or lenses.

Key points about circle intersections:

• Two circles can intersect in zero, one, or two points, depending on their radii and the distance between their centers.
• When they intersect, as in this exercise, they form two intersection points, and the arcs between them define a region.
• Calculating the area of intersection often requires determining the lengths of arcs and the shapes they form with chordal segments.

In our problem, the intersection involves determining the segments of the circle within each other's boundaries, using the calculated angles and understanding the geometric setup. It involves detailed work to calculate the exact area of the lune, which results from subtracting one segment's area from another. This intersection analysis is crucial, as the shape and size of the resulting lune is dependent entirely on how the circles overlap in geometry.