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Mathematical Excursions (MindTap Course List)4th Edition
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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.

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Mathematical Excursions (MindTap C...

4th Edition
Richard N. Aufmann + 3 others
Publisher: Cengage Learning
ISBN: 9781305965584

Solutions

Chapter
Section
Problem 1EE
Problem 2EE
Problem 3EE
Problem 4EE
Problem 5EE
Problem 6EE
Problem 1ES
Problem 2ES
Problem 3ES
Problem 4ES
Problem 5ES
Problem 6ES
Problem 7ES
Problem 8ES
Problem 9ES
Problem 10ES
Problem 11ES
Problem 12ES
Problem 13ES
Problem 14ES
Problem 15ES
Problem 16ES
Problem 17ES
Problem 18ES
Problem 19ES
Problem 20ES
Problem 21ES
Problem 22ES
Problem 23ES
Problem 24ES
Problem 25ES
Problem 26ES
Problem 27ES
Problem 28ES
Problem 29ES
Problem 30ES
Problem 31ES
Problem 32ES
Problem 33ES
Problem 34ES
Problem 35ES
Problem 36ES
Problem 37ES
Problem 38ES
Problem 39ES
Problem 40ES
Problem 41ES
Problem 42ES
Problem 43ES
Problem 44ES
Problem 45ES
Problem 46ES
Problem 47ES
Problem 48ES
Problem 49ES
Problem 50ES
Problem 51ES
Problem 52ES
Problem 53ES
Problem 54ES
Problem 55ES
Problem 56ES
Problem 57ES
Problem 58ES
Problem 59ES
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Mathematical Excursions (MindTap C...

4th Edition
Richard N. Aufmann + 3 others
Publisher: Cengage Learning
ISBN: 9781305965584
Chapter 7.3, Problem 58ES
Textbook Problem
2 views

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

Chapter 7.3, Problem 58ES, A circle with radius r and circumference C is sliced into 16 identical sectors which are then , example 1

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

Chapter 7.3, Problem 58ES, A circle with radius r and circumference C is sliced into 16 identical sectors which are then , example 2

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

  1. What dimension of the circle approximates the height of the parallelogram?
  2. What dimension of the circle approximates the base of the parallelogram?
  3. 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.

Explanation of Solution

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.

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Mathematical Excursions (MindTap Course List)
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Ch. 7.1 - Draw the yin-and-yang symbol shown at the right....Ch. 7.1 - Draw a hexagon with all sides the same length....Ch. 7.1 - Find the midpoint of the line segment AB.Ch. 7.1 - Draw the heart-shaped figure shown at the right....Ch. 7.1 - Convert between the two measurements. 6ft = in.Ch. 7.1 - Convert between the two measurements. 7920 ft = miCh. 7.1 - Convert between the two measurements. 5yd = in.Ch. 7.1 - Convert between the two measurements. 12 mi = ydCh. 7.1 - Convert between the two measurements. 64 oz = lbCh. 7.1 - Convert between the two measurements. 9000 lb =...
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For calculations...Ch. 7.5 - Find the volume of the figure. For calculations...Ch. 7.5 - Find the volume of the figure. For calculations...Ch. 7.5 - Find the volume of the figure. For calculations...Ch. 7.5 - Find the volume of the figure. For calculations...Ch. 7.5 - Find the volume of the figure. For calculations...Ch. 7.5 - Find the surface area of the figure. For...Ch. 7.5 - Find the surface area of the figure. For...Ch. 7.5 - Find the surface area of the figure. For...Ch. 7.5 - Find the surface area of the figure. For...Ch. 7.5 - Find the surface area of the figure. For...Ch. 7.5 - Find the surface area of the figure. For...Ch. 7.5 - Solve. Volume A rectangular solid has a length of...Ch. 7.5 - Solve. Volume Find the volume of a rectangular...Ch. 7.5 - Solve. Volume Find the volume of a cube whose side...Ch. 7.5 - Solve. Volume The length of a side of a cube is 7...Ch. 7.5 - Solve. Volume The diameter of a sphere is 6 ft....Ch. 7.5 - Solve. Volume Find the volume of a sphere that has...Ch. 7.5 - Solve. Volume The diameter of the base of a...Ch. 7.5 - Solve. Volume The height of a cylinder is 7.2 m....Ch. 7.5 - Solve. Volume The radius of the base of a cone is...Ch. 7.5 - Solve. Volume The height of a cone is 15 cm. The...Ch. 7.5 - Solve. Volume The length of a side of the base of...Ch. 7.5 - Solve. Volume The height of a regular square...Ch. 7.5 - Solve. The length of a side of a cube is equal to...Ch. 7.5 - Solve. A sphere and a cylinder have the same...Ch. 7.5 - Solve. The Statue of Liberty The index finger of...Ch. 7.5 - Solve. Fish Hatchery A rectangular tank at a fish...Ch. 7.5 - Solve. The Panama Canal When the lock is full, the...Ch. 7.5 - Solve. Surface Area The width of a rectangular...Ch. 7.5 - Solve. Surface Area The side of a cube measures...Ch. 7.5 - Solve. Surface Area Find the surface area of a...Ch. 7.5 - Solve. Surface Area Find the exact surface area of...Ch. 7.5 - Solve. Surface Area The radius of a sphere is 2...Ch. 7.5 - Solve. Surface Area The radius of the base of a...Ch. 7.5 - Solve. Surface Area The diameter of the base of a...Ch. 7.5 - Solve. Surface Area The slant height of a cone is...Ch. 7.5 - Solve. Surface Area The diameter of the base of a...Ch. 7.5 - Solve. Surface Area The length of a side of the...Ch. 7.5 - Solve. Surface Area The slant height of a regular...Ch. 7.5 - Solve. Appliances The volume of a freezer that is...Ch. 7.5 - Solve. Aquariums The length of a rectangular solid...Ch. 7.5 - Solve. Paint A can of paint will cover 300 ft2 of...Ch. 7.5 - Solve. Ballooning A hot air balloon is in the...Ch. 7.5 - Solve. Surface Area The length of a side of the...Ch. 7.5 - Find the volume of the figure. Round to the...Ch. 7.5 - Find the volume of the figure. Round to the...Ch. 7.5 - Find the volume of the figure. Round to the...Ch. 7.5 - Find the volume of the figure. Round to the...Ch. 7.5 - Find the volume of the figure. 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Round to the nearest tenth of a...Ch. 7.6 - Use a calculator. Round to the nearest tenth of a...Ch. 7.6 - Use a calculator. Round to the nearest tenth of a...Ch. 7.6 - Use a calculator. Round to the nearest tenth of a...Ch. 7.6 - Use a calculator. Round to the nearest tenth of a...Ch. 7.6 - Use a calculator. Round to the nearest tenth of a...Ch. 7.6 - Use a calculator. Round to the nearest tenth of a...Ch. 7.6 - Use a calculator. Round to the nearest tenth of a...Ch. 7.6 - Use a calculator. Round to the nearest tenth of a...Ch. 7.6 - Use a calculator. Round to the nearest tenth of a...Ch. 7.6 - Use a calculator. Round to the nearest tenth of a...Ch. 7.6 - Use a calculator. Round to the nearest tenth of a...Ch. 7.6 - Use a calculator. Round to the nearest tenth of a...Ch. 7.6 - Use a calculator. Round to the nearest tenth of a...Ch. 7.6 - Use a calculator. Round to the nearest tenth of a...Ch. 7.6 - Use a calculator. Round to the nearest tenth of a...Ch. 7.6 - Draw a picture and label it. Then set up an...Ch. 7.6 - Draw a picture and label it. Then set up an...Ch. 7.6 - Draw a picture and label it. Then set up an...Ch. 7.6 - Draw a picture and label it. Then set up an...Ch. 7.6 - Draw a picture and label it. Then set up an...Ch. 7.6 - Draw a picture and label it. Then set up an...Ch. 7.6 - Draw a picture and label it. Then set up an...Ch. 7.6 - Draw a picture and label it. Then set up an...Ch. 7.6 - Draw a picture and label it. Then set up an...Ch. 7.6 - Draw a picture and label it. Then set up an...Ch. 7.6 - Draw a picture and label it. Then set up an...Ch. 7.6 - Draw a picture and label it. Then set up an...Ch. 7.6 - Draw a picture and label it. Then set up an...Ch. 7.6 - Draw a picture and label it. 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 Lévys 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 32° angle.Ch. 7 - Find the volume of a rectangular solid with a...Ch. 7 - Two angles of a triangle measure 37° and 48°....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(–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 32° 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 40° 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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