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nature of mathematics

Questions and Answers of Nature Of Mathematics

Which of the figures in Problems 21–24 are Saccheri quadrilaterals?
Which of the figures in Problems 21–24 are Saccheri quadrilaterals?
Which of the figures in Problems 21–24 are Saccheri quadrilaterals?
Which of the figures in Problems 21–24 are Saccheri quadrilaterals?
Consider the cubes given in Problems 25–28, and draw the projections on the top, front, and side of each cube, as shown in Figure 7.75.Figure 7.75 1 TOP FRONT I | SIDE
Consider the cubes given in Problems 25–28, and draw the projections on the top, front, and side of each cube, as shown in Figure7.75.Figure 7.75 1 TOP FRONT I | SIDE
Consider the cubes given in Problems 25–28, and draw the projections on the top, front, and side of each cube, as shown in Figure 7.75.Figure 7.75 1 TOP FRONT I | SIDE
Consider the cubes given in Problems 25–28, and draw the projections on the top, front, and side of each cube, as shown in Figure 7.75.Figure 7.75 1 TOP FRONT I | SIDE
Consider the top, front, and side views as shown in Problems 29–32, and draw a cube similar to the one shown in Figure 7.75.Figure 7.75 1 TOP FRONT I | SIDE
Consider the top, front, and side views as shown in Problems 29–32, and draw a cube similar to the one shown in Figure 7.75.Figure 7.75 1 TOP FRONT I | SIDE
Consider the top, front, and side views as shown in Problems 29–32, and draw a cube similar to the one shown in Figure 7.75.Figure 7.75 1 TOP FRONT I | SIDE
Consider the top, front, and side views as shown in Problems 29–32, and draw a cube similar to the one shown in Figure 7.75.Figure 7.75 1 TOP FRONT I | SIDE
a. What are the length and width of a standard index card? .b. Write the ratio of length to width as a decimal and compare it with π.
a. What are the length and width of a standard brick?b. Write the ratio of length to width as a decimal and compare it with π.
Use Figure 7.65 to find the requested measurements in Problems 35–36 using your own body as the model.Figure 7.65 D b d C D B C F A
Use Figure 7.65 to find the requested measurements in Problems 35–36 using your own body as the model.Figure 7.65 D b d C D B C F A
A plant grows for two months and then adds a new branch. Each new branch grows for two months and then adds another branch. After the second month, each branch adds a new branch every month. Assume
Leo Moser studied the effect that two face-to-face panes of glass have on light reflected through the panes. If a ray is unreflected, it has just one path through the glass. If it has one reflection,
If a window is to be 5 feet wide, how high should it be, to the nearest tenth of a foot, to be a golden rectangle?
If a canvas for a painting is 18 inches wide, how high should it be, to the nearest inch, to be in the divine proportion?
A photograph is to be printed on a rectangle in the divine proportion. If it is 9 cm high, how wide is it, to the nearest centimeter?
If the Parthenon is 60 ft high, what is its width, to the nearest foot, if we assume the building conforms to the golden ratio?
On a globe, locate San Francisco, Miami, and Detroit. Connect these cities with the shortest paths to form a triangle. Next, use a protractor to measure the angles. What is their sum?
On a globe, locate Tokyo, Seattle, and Honolulu. Connect these cities with the shortest paths to form a triangle. Next, use a protractor to measure the angles. What is their sum?
Consider the spheres shown in Figure 7.76.Figure 7.76a. Explain what is meant by a great circle on a sphere.b. Do one or both of the lettered curves in the figure at the right above seem to be
In Figure 7.76, the sphere at the left shows two lines. In Euclidean geometry, we know that two lines are either parallel or cross at exactly one point. Discuss this property in relation to what you
In Figure 7.76, the lines on the sphere at the right appear to be parallel. In Table 7.2, we saidthat there are no parallels on a sphere. What’s wrong with our reasoning?Figure 7.76 A P m e
Walking north or south on the earth is defined as walking along the meridians, and walking east or west is defined as walking along parallels. There are points on the surface of the earth from which
In Lobachevskian geometry, the sum of the measures of the angles of a triangle is less than 180°. Discuss the following statement: In Lobachevskian geometry, the sum of the measures of the angles of
In Euclidean geometry, if two triangles are similar, then they may or may not also be congruent. In Lobachevskian geometry, if two triangles are similar, do you think that they must also be congruent?
“When is an open book a closed book?” If the length-to-width ratio of a book remains unchanged when that book is opened, then that book is said to be in the librarian’s ratio.(Note that the
Does this book satisfy the librarian’s ratio (see Problem 55)?Data from Problem 55“When is an open book a closed book?” If the length-to-width ratio of a book remains unchanged when that book
a. Show that the solutions to the equationb. Is there a relationship between the decimal representations of these numbers? If so, explain. x-x-1=0 1- 5 2 c=1+5 and 7 2 are T
If you form a list of numbers where the first two numbers are ones, and then find successive terms (in order) by adding these two numbers to obtain the next, you will find 1, 1, 2, 3, 5, 8, 13,...
Find two additional points satisfying the conditions stated in Problem 52.Data from Problem 52Walking north or south on the earth is defined as walking along the meridians, and walking east or west
Escher’s work Circle Limit IV is a tessellation (see Section 9.3) based on Lobachevskian geometry.
If a 4-cm cube is painted green and then cut into 64 1-cm cubes, how many of those cubes will be painted ona. 4 sides?b. 3 sides?c. 2 sides?d. 1 side?e. 0 sides?
What do you see in the following illustration?
Which of the following ideas (if any) does the illustration for Problem 2 show: inversion, reflection, contraction, dilation, or rotation?Data from problem 2What do you see in the following
Which of the following pictures illustrates a reflection? 2 DHIO OHIO 17 5128 STAR KATS drin
Which of the pictures in Problem 4 is upside down?Data from problem 4Which of the following pictures illustrates a reflection? 2 DHIO OHIO 17 5128 STAR KATS drin
Which of the following illustrate at least one line of symmetry? If a figure illustrates symmetry, show a line of symmetry. If it does not, tell why. a. b.
Bourbon Palace (Figure 7.77) is home to the Assemblée Nationale, the French parliament’s lower house.a. What is a golden rectangle?b. Does it look to you like the front of this building forms a
If the width of Bourbon Palace is 280 ft, what is the height, assuming that the front forms a golden rectangle?
One method for estimating the capacity of a boat is to divide the product of the length and width of the boat by 15. Write a formula to represent this idea.
Suppose ,l1 and ,l2 are parallel lines. Classify the angle orpairs of angles shown Figure 7.78.Figure 7.78 a. 41 c. 22 and 24 b. 25 and 26 d. 22 e. 27 f. 41 and 23 g. Can you classify any of the
If an angle is 49°, then . . .a. . . . what is its complement?b. . . . what is its supplement?c. . . . if it is an angle in a right triangle, what is the size of the other acute angle?
If 3x + 20°, 2x - 40°, and x - 16° are angles of a triangle, then what is x?
If a box is 10 in. wide, what is its height (to the nearest inch) if we assume the dimensions are in a golden ratio?
In an isosceles right triangle, if the hypotenuse is 10 in., how long are the other legs?
Find the value of the given trigonometric ratios. Round your answers to four decimal places.a. sin 59°b. tan 0°c. cos 18°d. tan 82°
The world’s most powerful lighthouse is on the coast of Brittany, France, and is about 160 ft tall. Suppose you are in a boat just off the coast, as shown in Figure 7.79. Determine your distance
For the die shown in Figure 7.80, fill in the blanks in each of the given dice.Figure 7.80 BO E
What is a Saccheri quadrilateral?
Discuss why this quadrilateral leads to different kinds of geometries.
In Figure 7.81 there are eight square rooms making up a maze. Each square room has two walls that are mirrors and two walls that are open spaces. Identify the mirrored walls, and then solve the maze
At the beginning of this chapter, we included a picture of a fly on or in a cube, and we saw that figures can sometimes be ambiguous. Figure 7.82a shows a cube with a dot in the middle of each face.
Place a dollar bill across the top of two glasses that are at least 3.5 in. apart. Now, describe how you can place a quarter dollar in the middle of the dollar bill without having it fall.
What is a geodesic dome? What is a Buckminster Fuller dome? Use toothpicks, rolled newspapers, or PVC pipe to build a model.
Write a report on optical illusions.
Create an aestheometry design beginning with an angle.
Create an aestheometry design beginning with a circle.
Write a paper discussing the nature of an unsolved problem as compared with an impossible problem. As part of your paper, discuss the problems of trisecting an angle and squaring a circle.
The Historical Note in Problem Set 7.5 on page 362 asks, “Why did the Egyptians build the pyramids using a slant height angle of about 44° or 52°?” Write a paper answering this question.Data
Do some research on the length-to-width ratios of the packaging of common household items. Form some conclusions. Find some examples of the golden ratio in art. Do some research on dynamic symmetry.
In Example 2 of Section 7.6, we assumed the width of the Parthenon to be 101 ft and found the height to be 62.4 ft (assuming the golden ratio). If you worked Problem 7 in Problem Set 7.6, you assumed
What is the golden ratio? What is the silver ratio? Write a paper on the silver ratio.
Write a paper on perspective. How are three-dimensional objects represented in two dimensions?
Is there a “best” rectangle? If you like to do origami, which size paper best suits your needs? What are “standard” sizes for paper?
The discovery and acceptance of non-Euclidean geometries had an impact on all of our thinking about the nature of scientific truth. Can we ever know truth in general? Write a paper on the nature of
Find the one composite number in the following set: 31 331 3331 33331 333331 3333331 33333331 333333331
Measure the given segment B a. to the nearest centimeter. b. to the nearest centimeter.
Find the perimeter of each polygon. a. b. d. 1 cm 1 cm 10 dm 1 cm 4 ft 10 dm 2 ft 10 dm C. e. 3 m 5 mi 5 mi 5 m 4 m
Suppose you have enough material for 70 ft of fence and want to build a rectangular pen 14 ft wide. What is the length of this pen?
Find the distance around each figure. a. 4 ft d. of a circle 2 dm b. e. 100 28 cm of a circle 30 cm C. 5 m
Contrast precision and accuracy.
What do we mean by length? What are the standard units of length? Arrange the units of length (both U.S. and metric) in size from smallest to largest.
State the perimeter formulas for a square, rectangle, equilateral triangle, and regular pentagon.
What is the agreement about the accuracy of answers in this text?
What is the formula for the circumference of a circle?
Define π .
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What is the area of the shaded region? a. 21 cm 15 cm b. 12 in. 12 in.
How many square feet (see Figure 8.9) are there in a square yard?Figure 8.9 1 yd 1 yd
Find the area (to the nearest tenth unit) of each shaded region. a. r=9.0 yd b. d = 10.0 m
Suppose your living room is 12 ft by 15 ft and you want to know how many square yards of carpet you need to cover this area.
How many acres are there in a rectangular property measuring 363 ft by 180 ft?
What do we mean by area?
How do you find the area of a circle? Contrast with the procedure for finding the circumference of a circle.
Estimate the area of each figure in Problems 3–8 to the nearest square centimeter. . b.
Estimate the area of each figure in Problems 3–8 to the nearest square centimeter. a. b.
Estimate the area of each figure in Problems 3–8 to the nearest square centimeter.
Estimate the area of each figure in Problems 3–8 to the nearest square centimeter.
Estimate the area of each figure in Problems 3–8 to the nearest square centimeter.
Estimate the area of each figure in Problems 3–8 to the nearest square centimeter.
Pick the best choices in Problems 9–18 by estimating. Do not measure. For metric measurements, do not attempt to convert to the U.S. system.The area of a dollar bill is A. 18 in. B. 6 in.2 C. 18
Pick the best choices in Problems 9–18 by estimating. Do not measure. For metric measurements, do not attempt to convert to the U.S. system.The area of a five-dollar bill isA. 100 cmB. 10

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