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nature of mathematics

Questions and Answers of Nature Of Mathematics

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 the front cover of this textbook isA. 80
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 VISA credit card isA. 8 in.B. 8 in.2C. 8
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 sheet of notebook paper isA. 90 cm2B. 10
Find the amount of paint needed for a box with edges 3 ft. 3 ft 3 ft 3 ft
Find the outside surface area. a. 50 cm 30 cm 80 cm b. 3 cm h 6 cm
You want to paint 200 ft of a three-rail fence, and you need to know how much paint to purchase.
Find the volume of a box that measures 4 ft by 6 ft by 4 ft.
Find the volume of each solid. a. 10 cm 10 cm 10 cm b. 10 cm 25 cm 4 cm 7 in. 11 in. 3 in.
Suppose you are pouring a rectangular driveway with dimensions 24 ft by 65 ft. The depth of the driveway is 3 in. and concrete is ordered by the yard. By a “yard” of concrete, we mean a cubic
How much water would each of the following containers hold? a. 90 cm 80 cm 40 cm b. 7 in. 22 in. 6 in.
An ecology swimming pool is advertised as being 20 ft × 25 ft × 5 ft. How many gallons will it hold?
Contrast length, area, and volume.
What do we mean by surface area?
Contrast volume and capacity.
Compare the sizes of a cubic inch and a cubic centimeter.
Compare the sizes of a quart and a liter.
Compare a meter and a yard.
In Problems 7–8, find the volume of each solid by counting the number of cubic centimeters in each box.
Find the volume of each solid in Problems 9–16. 5 ft 5 ft 5 ft
Find the volume of each solid in Problems 9–16. 12 in. 12 in. 12 in.
Find the volume of each solid in Problems 9–16. 20 cm 20 cm 20 cm
Find the volume of each solid in Problems 9–16. 1 yd 1 yd 1 yd
Find the volume of each solid in Problems 9–16. 2 ft 3 ft 4 ft
Find the volume of the fish tank shown in Figure 8.22 to the nearest cubic unit.Figure 8.22 18.00 18 AMAT Reuters/Landov
Find the volume of the Toblerone chocolate box shown in Figure 8.23 to the nearest tenth cubic inch. The base is an equilateral triangle with base 1 in. and height 1/2√3 in. The length of the box
Find the volume of the solid shown in Figure 8.24 to the nearest cubic unit. CAnton Ivanov/Shutterstock.com 5 m 5 m
Find the surface area and volume of the 12-in. globe shown in Figure 8.25. Give your answers to the nearest whole unit, paying attention to the unit.Figure 8.25 Oleg Lopatkin/Shutterstock
Find the surface area (to the nearest square unit), and also find the volume (to the nearest cubic unit) of the glass shown in Figure 8.26. The diameter of the glass is 5 cm and the height is 6
a. Compare the area of a square whose side is tripled with the area of the original square.b. Compare the area of a circle after its radius is doubled.c. Compare the volume of a sphere after its
Write each metric measurement using each of the other prefixes.a. 43 kmb. 60 Lc. 14.1 cg
Make the indicated conversions.a. 287 cm to kmb. 1.5 kL to Lc. 4.8 kg to g
List the measures for all six angles for the figures given in Problems 9–14. D 20 60 F E D' E' F'
Use the illustration in Figure 7.12 to draw the figures requested in Problems 11–19.Figure 7.12 RS
This problem anticipates a major result derived in the next section. Show that the sum of the measures of the interior angles of any triangle is 180°.a. Draw three triangles, one with all acute
Contrast congruent and similar triangles.
What does it mean when we say the corresponding sides of two similar triangles are proportional?
In Problems 3–8, determine if the triangles are similar. A b 20 C a F e BD d f 20 E
In Problems 3–8, determine if the triangles are similar. A' b' 20 c' F' a' 'B' D' d' f' 20 E'
In Problems 3–8, determine if the triangles are similar. b" 20 c" C" 68 a" F" e" B" D" 70 d" 20 E"
In Problems 3–8, determine if the triangles are similar. G h 70 8 60 H k 1 60 70 L j K
In Problems 3–8, determine if the triangles are similar. M n 60 P P m N S r q S 30 R
In Problems 3–8, determine if the triangles are similar. T V 45 U U 1 45 V W X W 45 Y
List the measures for all six angles for the figures given in Problems 9–14. A 75 ' 25 B' '
List the measures for all six angles for the figures given inProblems9–14. G 110 I H K L 32 J
List the measures for all six angles for the figures given in Problems 9–14. D 38 E 68 F A B C
List the measures for all six angles for the figures given in Problems 9–14. A C 44 B B' A' C'
List the measures for all six angles for the figures given in Problems 9–14. C A B' A' 36 C'
List the lengths of all six sides for the figures given in Problems 15–20. A 11 11 B 5 CA'. 22 B' C'
List the lengths of all six sides for the figures given in Problems 15–20. D 12 16 F 8 E H 22 G
List the lengths of all six sides for the figures given in Problems 15–20. D E 20 F G 14 H 16 12
List the lengths of all six sides for the figures given in Problems 15–20. A B' 15 5 3 B A' 4 C
List the lengths of all six sides for the figures given in Problems 15–20. 8 6 B A 10 B' A' 5 C'
List the lengths of all six sides for the figures given in Problems 15–20. D 9 F E D' 17 6 E' 15 'F'
Given two similar triangles, as shown in Figure 7.46, find the unknown lengths in Problems 21–28.a = 4, b = 8; find c.Figure 7.46 B a C b A B' a' C' c' b' 'A'
Given two similar triangles, as shown in Figure 7.46, find the unknown lengths in Problems 21–28.a' = 7, b' = 3; find c'.
Given two similar triangles, as shown in Figure 7. 46, find the unknown lengths in Problems 21–28.a = 4, b = 8, a' = 2; find b'.Figure 7.46 B a C b A B' a' C' c' b' 'A'
Given two similar triangles, as shown in Figure 7.46, find the unknown lengths in Problems 21–28.b = 5, c = 15, b' = 3; find c'.Figure 7.46 B a C b A B' a' C' c' b' 'A'
Given two similar triangles, as shown in Figure 7.46, find the unknown lengths in Problems 21–28.c = 6, a = 4, c' = 8; find a'.Figure 7.46 B a C b A B' a' C' c' b' 'A'
Given two similar triangles, as shown in Figure 7. 46, find the unknown lengths in Problems 21–28.a' = 7, b' = 3, a = 5; find b.Figure 7.46 B a C b A B' a' C' c' b' 'A'
Given two similar triangles, as shown in Figure 7.46, find the unknown lengths in Problems 21–28.b' = 8, c' = 12, c = 4; find b.Figure 7.46 B a C b A B' a' C' c' b' 'A'
Given two similar triangles, as shown in Figure 7.46, find the unknown lengths in Problems 21–28.c' = 9, a' = 2, c = 5; find a.Figure 7.46 B a C b A B' a' C' c' b' 'A'
Each figure in Problems 29–34 contains two similar triangles. Find the unknown measure indicated by a variable. Answer to the nearest tenth. h 30 10 25 11
Each figure in Problems 29–34 contains two similar triangles. Find the unknown measure indicated by a variable. Answer to the nearest tenth. 40 20 60 h
Each figure in Problems 29–34 contains two similar triangles. Find the unknown measure indicated by a variable. Answer to the nearest tenth. 10 5 X 18
Each figure in Problems 29–34 contains two similar triangles. Find the unknown measure indicated by a variable. Answer to the nearest tenth. 5 X 2 3
Each figure in Problems 29–34 contains two similar triangles. Find the unknown measure indicated by a variable. Answer to the nearest tenth. 20 55 y 30
Each figure in Problems 29–34 contains two similar triangles. Find the unknown measure indicated by a variable. Answer to the nearest tenth. 15 14 N 20
In the 1960s, a fast-food chain called Der Wienerschnitzel had outlets built in a triangular shape. Classify the type of triangle that is used for the front of the building shown in
The Vanke Triple V Gallery by the Ministry of Design in Tianjin, China, has an interesting triangular design, as shown in Figure 7.48. The width of the triangle in the front is 2/3 of the height of
Given A̅C̅ is perpendicular to M̅B̅, and ΔABC is an equilateral triangle. Show that ΔABM ∼ ΔCBM. A B M C
Given m∠D = m∠E. Show that ΔABD ~ ΔCBE. A D B E C
Given that C̅T̅ bisects both ∠ACO and ∠ATO. Show that ΔCAT ΔCOT. A O T
Given that A̅W̅ bisects ∠R1AR2 and ∠R1,OR2. Show that ΔAOR1 ∼ ΔAOR2. A O R R W
Use similar triangles and a proportion to find the length of the lake shown in Figure 7.49.Figure 7.49 K 210 ft 50.0 ft -140 ft P
Suppose the distances in Problem 41 are changed as follows: 150 ft instead of 210 ft and 90 ft instead of 140 ft. What is the length of this lake (to the nearest foot)?
Use similar triangles and a proportion to find the height of the house siding shown in Figure 7.50.Figure 7.50 A 4 ft 6 ft 16 ft 8.
Suppose the 6 ft distance in Problem 43 is 5 ft 8 in. Use this information to find the height of the house (to the nearest inch).
A building casts a shadow 75 ft long. At the same time, the shadow cast by a vertical yardstick is 5 ft long. How tall is the building?
Suppose the shadow of the building in Problem 45 is 75 ft 3 in. How tall is the building (to the nearest inch)?
A bell tower casts a shadow 45 ft long. At the same time, the shadow cast by a vertical yardstick is 23 in. long. How tall is the bell tower (to the nearest foot)?
A building casts a shadow 23 ft long. At the same time, the shadow cast by a 6 ft person is 134 ft. How tall is the building (to the nearest foot)?
If a tree casts a shadow of 12 ft at the same time that a 6-ft person casts a shadow of 2 1/2 ft, find the height of the tree (to the nearest foot).
If a tree casts a shadow of 8 ft 3 in. at the same time that a 5-ft 10-in. person casts a shadow of 2 ft 7 in., find the height of the tree (to the nearest inch).
If a tree casts a shadow of 4 ft 5 in. at the same time that a 5-ft 9-in. person casts a shadow of 3 ft 10 in., find the height of the tree (to the nearest inch).
If lines are drawn on a map, a triangle can be formed by the cities of New York City, Washington, D.C., and Buffalo, New York. On the map, the distance between New York City and Washington, D.C., is
If lines are drawn on a map, a triangle can be formed by the cities of New Orleans, Louisiana; Denver, Colorado; and Chicago, Illinois. On the map, the distance between New Orleans and Denver is 10.
Suppose a 6-ft person wishes to determine the height of a bridge above the bottom of a canyon, as shown in Figure 7.51.Figure 7.51To do this, this person stands at one end of the bridge (point A)
Suppose a 6-ft person wishes to determine the height of a footbridgeconnecting two buildings, as shown in Figure 7.52.Figure 7.52To do this, this person stands at one end of the bridge (point S)
A useful theorem that uses proportionality, but not triangles, is the following: If two lines connect the endpoints of parallel segments of different lengths, then a line through the intersection of
Divide the segment given in Problem 57 into two parts in a 3-to-7 ratio.Data from problem 57A useful theorem that uses proportionality, but not triangles, is the following: If two lines connect the
Present an argument showing that if two triangles are equilateral, then they are similar triangles.
For any right triangle ABC (right angle at C), drop a perpendicular from point C to base A̅B̅ at the point D. Show that the two triangles thus formed are similar.
Find the cosine, sine, and tangent of \(45^{\circ}\).
Find the trigonometric ratios by using a calculator. Round your answers to four decimal places.a. \(\sin 45^{\circ}\)b. \(\cos 32^{\circ}\)c. \(\tan 19^{\circ}\)
The angle from the ground to the top of the Great Pyramid of Cheops is \(52^{\circ}\) if a point on the ground directly below the top is \(351 \mathrm{ft}\) away. (See Figure 7.55.) What is the
Given a right triangle with sides of length 5 and 12, find the measures of the angles of this triangle.
The angle of elevation to the top of a tree from a point on the ground \(42 \mathrm{ft}\) from its base is \(33^{\circ}\). Find the height of the tree (to the nearest foot).
What is a sine?
The mathematics of the early Egyptians was practical and centered around surveying, construction, and recordkeeping. They used a simple device to aid surveying - a rope with 12 equal divisions marked
Use the right triangle in Figure 7.57 to answer the questions in Problems 6-16. Figure 7. 57What is the side opposite ∠A∠A ? A b a B

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