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4th Edition

Richard N. Aufmann + 3 others

Publisher: Cengage Learning

ISBN: 9781305965584

Chapter 7.3, Problem 58ES

Textbook Problem

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A circle with radius r and circumference C is sliced into 16 identical sectors which are then arranged as shown below.

The exterior of this figure resembles a parallelogram, and the figure has the same area as the circle. |

The exterior of the figure shown by the rearranged sectors resembles a parallelogram.

- What dimension of the circle approximates the height of the parallelogram?
- What dimension of the circle approximates the base of the parallelogram?
- Explain how the formula for the area of a circle can be derived by using this slicing approach.

To determine

**(a)**

To determine the dimension of the circle that approximates the height of the parallelogram.

**Given:**

Given, a circle with radius r and circumference C is sliced into 16 identical sectors which are then arranged as shown below:

Also, the exterior of the figure shown by the rearranged sectors resembles a parallelogram.

To determine

**(b)**

To determine the dimension of the circle that approximates the base of the parallelogram.

To determine

**(c)**

**To explain:**

To explain how the formula for the area of a circle can be derived by using the slicing approach.

Mathematical Excursions (MindTap Course List)

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Then set up an...Ch. 7.6 - As we noted in this section, angles can also be...Ch. 7.6 - As we noted in this section, angles can also be...Ch. 7.6 - As we noted in this section, angles can also be...Ch. 7.6 - As we noted in this section, angles can also be...Ch. 7.6 - Recall that the circumference of a circle is given...Ch. 7.6 - Recall that the circumference of a circle is given...Ch. 7.6 - Convert degree measure to radian measure. Find an...Ch. 7.6 - Convert degree measure to radian measure. Find an...Ch. 7.6 - Convert degree measure to radian measure. Find an...Ch. 7.6 - Convert degree measure to radian measure. Find an...Ch. 7.6 - Convert degree measure to radian measure. Find an...Ch. 7.6 - Convert degree measure to radian measure. Find an...Ch. 7.6 - Convert radian measure to degree measure. For...Ch. 7.6 - Convert radian measure to degree measure. For...Ch. 7.6 - Convert radian measure to degree measure. For...Ch. 7.6 - Convert radian measure to degree measure. For...Ch. 7.6 - Convert radian measure to degree measure. For...Ch. 7.6 - Convert radian measure to degree measure. For...Ch. 7.7 - Place two points A and B on a paper cup so that A...Ch. 7.7 - The only geodesics of a sphere are great circles....Ch. 7.7 - Write a sentence that explains how you know that...Ch. 7.7 - Excursion Exercises 4 to 6 require a world globe...Ch. 7.7 - Excursion Exercises 4 to 6 require a world globe...Ch. 7.7 - Excursion Exercises 4 to 6 require a world globe...Ch. 7.7 - The yellow surface at the left above is called a...Ch. 7.7 - State the parallel postulate for each of the...Ch. 7.7 - Name the mathematician who is called the...Ch. 7.7 - Name the mathematician who was the first to...Ch. 7.7 - What is the maximum number of right angles a...Ch. 7.7 - What name did Lobachevsky give to the geometry...Ch. 7.7 - Explain why great circles in Riemannian geometry...Ch. 7.7 - What is a geodesic?Ch. 7.7 - In which geometry can two distinct lines be...Ch. 7.7 - What model was used in this text to illustrate...Ch. 7.7 - In which geometry do all perpendiculars to a given...Ch. 7.7 - Find the exact area of a spherical triangle with...Ch. 7.7 - Find the area of a spherical triangle with three...Ch. 7.7 - City Geometry Find the Euclidean distance between...Ch. 7.7 - City Geometry Find the Euclidean distance between...Ch. 7.7 - City Geometry Find the Euclidean distance between...Ch. 7.7 - City Geometry Find the Euclidean distance between...Ch. 7.7 - City Geometry Find the Euclidean distance between...Ch. 7.7 - City Geometry Find the Euclidean distance between...Ch. 7.7 - City Geometry Find the Euclidean distance between...Ch. 7.7 - City Geometry Find the Euclidean distance between...Ch. 7.7 - A Distance Conversion Formula The following...Ch. 7.7 - A Distance Conversion Formula The following...Ch. 7.7 - A Distance Conversion Formula The following...Ch. 7.7 - A Distance Conversion Formula The following...Ch. 7.7 - A Distance Conversion Formula The following...Ch. 7.7 - A Distance Conversion Formula The following...Ch. 7.7 - Explain why there is no formula that can be used...Ch. 7.7 - a. If dE(P,Q)=dE(R,S), must dC(P,Q)=dC(R,S)?...Ch. 7.7 - Plot the points in the city circle with center...Ch. 7.7 - Plot the points in the city circle with center...Ch. 7.7 - Plot the points in the city circle with center (0,...Ch. 7.7 - Plot the points in the city circle with center (0,...Ch. 7.7 - How many points are on the city circle with center...Ch. 7.7 - Which of the following city circles has the most...Ch. 7.7 - Apartment Hunting in a City Use the following...Ch. 7.7 - Apartment Hunting in a City Use the following...Ch. 7.7 - A Finite Geometry Consider a finite geometry with...Ch. 7.7 - A Finite Geometry Consider a finite geometry with...Ch. 7.8 - Use a strip of paper and the folding procedure...Ch. 7.8 - Use the left-right rule to draw stage 2 of the...Ch. 7.8 - Use the left-right rule to determine the direction...Ch. 7.8 - Use the left-right rule to determine the direction...Ch. 7.8 - Use the left-right rule to determine the direction...Ch. 7.8 - Use an iterative process to draw stage 2 and stage...Ch. 7.8 - Use an iterative process to draw stage 2 and stage...Ch. 7.8 - Use an iterative process to draw stage 2 of the...Ch. 7.8 - Use an iterative process to draw stage 2 of the...Ch. 7.8 - Use an iterative process to draw stage 2 of the...Ch. 7.8 - Use an iterative process to draw stage 2 of the...Ch. 7.8 - Use an iterative process to draw stage 2 of the...Ch. 7.8 - Use an iterative process to draw stage 2 of the...Ch. 7.8 - Draw stage 3 and stage 4 of the fractal defined by...Ch. 7.8 - Draw stage 3 and stage 4 of the fractal defined by...Ch. 7.8 - Compute, if possible, the similarity dimension of...Ch. 7.8 - Compute, if possible, the similarity dimension of...Ch. 7.8 - Compute, if possible, the similarity dimension of...Ch. 7.8 - Compute, if possible, the similarity dimension of...Ch. 7.8 - Compute, if possible, the similarity dimension of...Ch. 7.8 - Compute, if possible, the similarity dimension of...Ch. 7.8 - Compute, if possible, the similarity dimension of...Ch. 7.8 - Compute, if possible, the similarity dimension of...Ch. 7.8 - Compute, if possible, the similarity dimension of...Ch. 7.8 - Compute, if possible, the similarity dimension of...Ch. 7.8 - Compare Similarity Dimensions Rank, from largest...Ch. 7.8 - The Peano Curve The Peano curve is defined by the...Ch. 7.8 - Explain why the similarity dimension formula...Ch. 7.8 - Stage 0 and stage 1 of LĂ©vys curve (see Exercise...Ch. 7 - For Exercises 1 to 5, convert one measurement to...Ch. 7 - For Exercises 1 to 5, convert one measurement to...Ch. 7 - For Exercises 1 to 5, convert one measurement to...Ch. 7 - For Exercises 1 to 5, convert one measurement to...Ch. 7 - For Exercises 1 to 5, convert one measurement to...Ch. 7 - The price of a beverage is $3.56 per liter. What...Ch. 7 - Given that in ma=74 and in mb=52, find the...Ch. 7 - Triangles ABC and DEF are similar. Find the...Ch. 7 - Find the volume of the geometric solid.Ch. 7 - Find the measure of x.Ch. 7 - Find the surface area of the rectangular solid.Ch. 7 - The length of a diameter of the base of a cylinder...Ch. 7 - Given that BC=11 cm and that AB is three times the...Ch. 7 - Given that mx=150, find the measures of w and y.Ch. 7 - Find the area of a parallelogram that has a base...Ch. 7 - Find the volume of the square pyramid.Ch. 7 - Find the circumference of a circle that has a...Ch. 7 - Given that l1l2, find the measures of angles a and...Ch. 7 - Find the supplement of a 32Â° angle.Ch. 7 - Find the volume of a rectangular solid with a...Ch. 7 - Two angles of a triangle measure 37Â° and 48Â°....Ch. 7 - The height of a triangle is 7 cm. The area of the...Ch. 7 - Find the volume of a sphere that has a diameter of...Ch. 7 - Framing The perimeter of a square picture frame is...Ch. 7 - Paint A can of paint will cover 200 ft2 of...Ch. 7 - Parks and Recreation The length of a rectangular...Ch. 7 - Patios What is the area of a square patio that...Ch. 7 - Landscaping A walkway 2 m wide surrounds a...Ch. 7 - Determine whether the two triangles are congruent....Ch. 7 - Find the unknown side of the triangle. Round to...Ch. 7 - In Exercises 31 and 32, find the values of sin ,...Ch. 7 - In Exercises 31 and 32, find the values of sin ,...Ch. 7 - Use a calculator. Round to the nearest tenth of a...Ch. 7 - Use a calculator. Round to the nearest tenth of a...Ch. 7 - Use a calculator. Round to the nearest tenth of a...Ch. 7 - Use a calculator. Round to the nearest tenth of a...Ch. 7 - Surveying Find the distance across the marsh in...Ch. 7 - Angle of Depression The distance from a plane to a...Ch. 7 - Angle of Elevation The angle of elevation from a...Ch. 7 - What is another name for Riemannian geometry?Ch. 7 - What is another name for Lobachevskian geometry?Ch. 7 - Name a geometry in which the sum of the measures...Ch. 7 - Name a geometry in which there are no parallel...Ch. 7 - Determine the exact area of the spherical...Ch. 7 - Determine the exact area of the spherical...Ch. 7 - Euclidean and City Distances Find the Euclidean...Ch. 7 - Euclidean and City Distances Find the Euclidean...Ch. 7 - Euclidean and City Distances Find the Euclidean...Ch. 7 - Euclidean and City Distances Find the Euclidean...Ch. 7 - Consider the points P( 1, 1), Q(4, 5), and R(â€“4,...Ch. 7 - Draw stage 0, stage 1, and stage 2 of the Koch...Ch. 7 - Draw stage 2 of the fractal with the following...Ch. 7 - For the fractal defined in Exercise 52, determine...Ch. 7 - Compute the similarity dimension of a strictly...Ch. 7 - Find the volume of a cylinder with a height of 6 m...Ch. 7 - Find the perimeter of a rectangle that has a...Ch. 7 - Find the complement of a 32Â° angle.Ch. 7 - Find the area of a circle that has a diameter of 2...Ch. 7 - In the figure below, lines l1 and l2 are parallel....Ch. 7 - In the figure below, lines l1 and l2 are parallel....Ch. 7 - Convert: 1.2 m = ________ cm.Ch. 7 - Find the volume of the figure. Give the exact...Ch. 7 - Triangles ABC and DEF arc similar. Find side BC.Ch. 7 - A right triangle has a 40Â° angle. Find the...Ch. 7 - Find the measure of x.Ch. 7 - Find the area of the parallelogram shown below.Ch. 7 - Surveying Find the width of the canal shown in the...Ch. 7 - Pizza How much more area is in a pizza with radius...Ch. 7 - Determine whether the two triangles are congruent....Ch. 7 - For the right triangle shown below, determine the...Ch. 7 - Find the values of sin , cos , and tan for the...Ch. 7 - Angle of Elevation From a point 27 ft from the...Ch. 7 - Trees Find the cross-sectional area of a redwood...Ch. 7 - Toolbox A toolbox is 14 in. long, 9 in. wide, and...Ch. 7 - Find the speed, in meters per second, of a car...Ch. 7 - State the Euclidean parallel postulate.Ch. 7 - What is a great circle, and what formula is used...Ch. 7 - Find the area of a spherical triangle with a...Ch. 7 - City Geometry Find the Euclidean distance and the...Ch. 7 - City Geometry How many points are on the city...Ch. 7 - Draw stage 2 of the fractal with the given...Ch. 7 - Draw stage 2 of the fractal with the given...Ch. 7 - Compute the replacement ratio, scale ratio, and...Ch. 7 - Compute the replacement ratio, scale ratio, and...

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