Ch. 4 - Applications of the Derivative

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 4 - Applications of the Derivative

Problem 74b

Chapter 4, Problem 74b

The arbelos An arbelos is the region enclosed by three mutually tangent semicircles; it is the region inside the larger semicircle and outside the two smaller semicircles (see figure). <IMAGE>
b. Show that the area of the arbelos is the area of a circle whose diameter is the distance BD in the figure.

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Identify the semicircles: Consider the larger semicircle with diameter AC and two smaller semicircles with diameters AB and BC, where B is the point of tangency between the two smaller semicircles.

Calculate the area of the larger semicircle: The area of a semicircle is given by \( \frac{1}{2} \pi r^2 \), where \( r \) is the radius. For the larger semicircle, the radius is \( \frac{AC}{2} \).

Calculate the area of the two smaller semicircles: Similarly, calculate the areas of the semicircles with diameters AB and BC. The radii are \( \frac{AB}{2} \) and \( \frac{BC}{2} \) respectively.

Determine the area of the arbelos: Subtract the sum of the areas of the two smaller semicircles from the area of the larger semicircle to find the area of the arbelos.

Relate the area of the arbelos to a circle: Show that this area is equivalent to the area of a circle with diameter BD. Use the formula for the area of a circle \( \pi \left( \frac{d}{2} \right)^2 \), where \( d \) is the diameter, and verify that it matches the area of the arbelos.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Arbelos Definition

An arbelos is a geometric figure formed by three mutually tangent semicircles. It consists of one larger semicircle that encompasses two smaller semicircles, all sharing a common tangent point. Understanding the configuration of these semicircles is crucial for analyzing the area and properties of the arbelos.

Area of Semicircles

The area of a semicircle can be calculated using the formula A = (1/2)πr², where r is the radius. In the context of the arbelos, the areas of the individual semicircles contribute to the overall area of the arbelos. Recognizing how to compute these areas is essential for deriving the area of the arbelos.

Circle Area Relation

The area of a circle is given by the formula A = π(d/2)², where d is the diameter. In the problem, it is stated that the area of the arbelos can be expressed as the area of a circle with a diameter equal to the distance BD. Understanding this relationship allows for the comparison and calculation of areas between the arbelos and the circle.

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