y = k for some constant k. x Equivalently, y = kx where k is the constant of variation.

Transcription

1

2 Section 6. Variation Variation Two variable quantities are often closely linked; if you change one then the other also changes. For instance, two quantities x and y might satisfy the following properties: If you double x then y also doubles. If you triple x then y also triples. For example, when r is the radius of a circle and C is the circumference of the circle, the usual formulas for circles tell us that C = πr. Now, when r = 3 then C = π (3) = 6π (approx.). Doubling the radius to r = 6 we compute C = π (6) = π , which is double the circumference when the radius is 3. You can check for yourself, in general, that if the radius doubles then the circumference also doubles. This is an example of direct variation of two quantities. The statement y varies directly as x means that: if x is multiplied by some factor then y is multiplied by the same factor. Another way to express this is that the ratio of y to x doesn t change: it is constant. In formulas, this says y = k for some constant k. x Equivalently, y = kx where k is the constant of variation. As another example, suppose an object is moving. Let x be its speed measured in feet per second (ft/s) and let y be the speed in miles per hour (mi/h or mph). Since x and y both measure the same speed, we see that if one of them doubles (or triples) so does the other. That is a direct variation, so there should be a formula y = kx. Suppose that we know that 88 ft/s equals 60 mph. Plugging these values into the variation formula we find 60 = k 88 so k = =. Therefore, y = x for 88 every speed. For instance, let s convert 00 ft/s to miles per hour. From x = 00 5 ft/s, the formula implies that y = 00 = mph. Two variable quantities can also be inversely related: If you double x then y becomes half as large. If you triple x then y becomes one-third as large. For example, suppose that you go on a 00-mile trip, driving at a constant speed. If you go 50 mph, the trip will take hours. If you double the speed to 00 mph, the trip will take you one hour. This is half as long as before. If you halve the speed to 5 mph, the trip will take you 4 hours. This is twice as long as it would take traveling at 50 mph. If we let r be the speed (or rate ) and let t be the time for the trip, then the standard formula says that 00 = rt.

3 48 Chapter 6 Ratios If we multiply the rate by some factor c, then we must divide the time by the same factor c (or equivalently multiply by c ). This is an example of inverse variation of two quantities. The statement y varies inversely as x means that: if x multiplied by some factor then y is multiplied by the reciprocal of that factor. Another way to express this is that the product xy doesn t change: it is constant. In formulas, this says xy = k for some constant k. k Equivalently, y = where k is the constant of variation. x There are other versions of variation that are commonly used. One of these is called joint variation. With joint variation, one quantity varies directly as the product of two other quantities. You can also have a value that varies directly or inversely as the square of another quantity. For example if two objects in space are considered, the gravitational force they exert on each other varies jointly as their masses and inversely as the square of the distance between them. This is called Newton s Law of Universal Gravitation. In formula, if m and m represent the masses of the objects, d the distance between mm them, and F the force, then F = G, where G is the constant of variation. d Type of Variation Mathematical English Statement Statement Direct Variation y = kx y varies directly as x y is directly proportional to x Inverse Variation k y varies inversely as x y = x y is inversely proportional to x Joint Variation y = kxz y varies jointly as x and z In all cases, k is a constant and is called the constant of proportionality or the constant of variation. It is important to understand the terminology associated with the different types of variation. Translating the relationship from the English statements to the mathematical statements is often the most difficult part of a problem. The table above should be used as guide to help translate from English to Mathematics.

4 Section 6. Variation 49 Example : Several equations are given below. Tell the type of variation that each represents, find the constant of proportionality, and rewrite the mathematical statement as a sentence in English.. Solution t s = This is direct variation with k =. In English: s varies directly as t.. K 3 = This is inverse variation with k = 3. In English: K varies m inversely as m. 3. V = Ah 3 This is joint variation with k =. In English: V varies 3 jointly as A and h. Example : In each problem, find the constant of variation.. y varies directly as x. When y = 5, x = 5. Solution: Since y varies directly as x, we have that y = kx. From this formula and the given x and y, we have 5 = 5k or k = 3.. The distance (d) traveled by a falling object varies directly as the square of the time (t) after its release. When d = 64, t =. Solution: Since d varies directly as t, we have that d the given information for d and t, we have that 64 = k = 6. = kt k( ) =. Using 4k so Exercises 6. Mental Math.. h varies directly as j. If h changes from to 4, what will happen to j?. h varies inversely as j. If h changes from to 4, what will happen to j? 3. The relationship between minutes and seconds is a type of variation. What type of variaton is it? Write a mathematical formula that relates the two quantities and find the constant of variation.

5 430 Chapter 6 Ratios 4. The relationship between inches and miles is a type of variation. What type of variation is it? Write a mathematical formula that relates the two quantities and find the constant of variation. Write the formula that illustrates each situation. This includes finding a value for k. 5. y varies inversely as x: When y = 3, x = A varies jointly as l and w: When A = 40, l = 0, and w = A varies directly as the square of r: When A = , r = T varies jointly as m and n and inversely as t. When T = 54, m = 3, n =, and t =.5. Solve. 9. The volume V of a right circular cone varies jointly as the height of the cone and the square of the radius of its base. The constant of proportionality is π 3. Write an equation that relates the quantities. 0. The weight of an object on the planet Naboo varies directly with its weight on the planet Endor. An object that weighs 45 pounds (lb) on Naboo weighs 5 lb on Endor. How much would an object that weighs 90 lb on Naboo weigh on Endor? Wickett, the Ewok, weighs 60 lb on Endor. How much does he weigh on Naboo?. The stopping distance of a car varies directly as the square of its speed. If a car traveling 60 mph can stop in 00 ft, how fast can that car travel and still stop in 68 ft?. Hooke s law states that the distance d that a spring will stretch varies directly as the mass m hanging from the spring. If a 4-kg mass stretches the spring 50 cm, how far will a 3-kg mass stretch the spring? 3. The maximum number of grams of fat that should be in a diet varies directly as a person s weight. A person weighting 80 lb should have no more than 90 grams of fat per day. What is the maximum daily fat intake for a person weighing 00 lb? 4. The maximum safe load for a horizontal rectangular beam varies jointly with the width of the beam and the square of the thickness of the beam and inversely with its length. If an 8-foot beam will support up to 750 pounds when the beam is 4 inches wide and inches thick, what is the maximum safe load in a similar beam 0 feet long, 8 inches wide, and 4 inches thick? 5. The resistance (in ohms) of a circular conductor varies directly with the length of the conductor and inversely with the square of the radius of the 3 conductor. If 5 feet of wire with a radius of 3 0 inches has a resistance of 0 ohms, what would be the resistance of 50 feet of the same wire if the radius is increased to inches.

6 Section 6. Area and Arclength Area and Arclength of a Circular Sector Before starting our discussion of arclength and area of a sector, let s take a look at a few well-known facts about circles and angles. The area of a circle is given by the formula A = πr and the circumference is given by the formula C = πr, where r is the radius of the circle. (Given our recent discussion of variation, one could say that the area of a circle varies directly as the square of the radius and the circumference varies directly as the radius.) An angle is defined to be two rays, called sides, that meet at a common endpoint. This endpoint is called the vertex. The angle divides the plane into two regions, one that is in the interior of the angle and the other is in the exterior of the angle. We measure an angle in units called degrees (denoted ). We have special names for angles depending on their degree measure. An angle that measures between 0 and 90 is called an acute angle. An angle that measures 90 is called a right angle. An angle that measures between 90 and 80 is called an obtuse angle. An angle that measures 80 is called a straight angle. Two angles are said to be complementary if the sum of their measures is 90. Two angles are said to be supplementary if the sum of their measure is 80. Given these basic truths about circles and angles, we need to develop some terminology to use when discussing circular sectors. A central angle is an angle whose vertex is at the center of a circle. A central angle determines a sector of the circle (the pie-shaped region in the interior of that angle). The boundary of that sector consists of two radii (straight line segments from the center) and an arc of the circle. That arc is said to be subtended by the central angle. In Figure, s is the arc subtended by the central angle θ. θ r Figure s Some rather obvious questions arise when we think about the sector of a circle determined by a central angle θ. What is the length of the circular arc (arclength) bounding this sector? What is the area of this sector? (We will denote the area of a sector as Area s and the area of the circle as Area c.) How are these affected by the size of the angle or by the radius of the circle? These questions can be answered by using ratios.

7 43 Chapter 6 Ratios Example θ Figure To summarize, we have the following results: Let s take a look at a simple example and determine the sector area and the arclength. In Figure 3, the central angle θ subtends exactly half of the circle. This angle θ measures 80, exactly half of the 360 for the full circle. Given the radius r, we know the area of the entire circle is πr. The area for that sector should be half the area of the entire circle. Likewise, the length of the arc should be half of the circumference of the circle. Remember, that the circumference is πr. = 360 = 80 s c Arclength = Circumference = ( πr ) θ Area = Area = ( πr ) Therefore, the angle, sector area, and arclength are all obtained by multiplying the full circle values by. Let s examine another circle and use ratios to relate the central angle, the sector area, and the arclength. Example θ Figure 3 In this example the central angle θ subtends onefourth of the circle. Using the same reasoning as in Example, the angle measures one-fourth of 360, or 90. The area of the sector is one-fourth the area of the circle and the arclength is one-fourth the circumference of the circle. In symbols: θ = 360 = 90 4 Area s = Areac = ( πr ) 4 4 Arclength = Circumference = ( π r) 4 4 With a bit of algebra we can express these observations in terms of ratios. In Example we had:

8 Section 6. Area and Arclength 433 θ 360 = Area Area s c = Arclength = Circumference In Example we had: θ 360 = 4 Area Area s c = 4 Arclength = Circumference 4 These examples motivate the general situation for any central angle θ in a circle: θ Areas Arclength = = 360 Area Circumference c The ratio of the area of the sector to the area of the circle equals the ratio of the measure of the central angle to 360. Similarly, the ratio of the arclength and the circumference equals the same ratio of the angles. Please note that the ratios that we get are unitless values since the units (like square inches or degrees) will cancel when dividing. Because of this, it is important that the numerator and denominator of each ratio have the same units. These ratios often provide enough information to compute areas of sectors and lengths of circular arcs. For each situation, we need to choose the appropriate pair of ratios based on the given information. Example 3: Find the area and the arclength of a sector in a circle with a radius of 3 inches if the central angle is 40. Areas θ Arclength θ Solution: Using the fact that = and =, we have Areac 360 Circumference 360 that Area s = (3) Area c = π = 9π = π sq in and Arclength = Circumference = π (3) = 6π (Bonus Question: 40 is what fraction of the entire circle?) in Example 4: Find the arclength of a sector in a circle with radius 5.5 feet if the central angle is 35. Solution: Since 35 is larger than 80, the arclength will be more than half of the circumference. So,

9 434 Chapter 6 Ratios Arclength = Circumference = π (5.5) feet Example 5: Suppose a circular sector has an area of 4 square inches. What is the measure of the central angle if the area of the circle is 36 square inches? Solution: Well, to answer this, let us look at the ratio that we get when comparing the area of the sector to the area of the whole circle. The area of the sector is 4 sq 4 7 in and the area of the circle is 36 sq in so that gives us a ratio of = and using 36 8 Areas θ 7 the fact that =, we get that θ = 360 = 40. Area c Similar thinking can be applied when dealing with objects that are rotating around a fixed point. The idea to remember is that one revolution of the object is one time around the entire circle. So one revolution corresponds to 360. The distance traveled in one revolution by an object on the edge of this circle would then be the same as the circumference of the circle. Example 6: Suppose that a pulley with a 6-inch diameter makes one and a half revolutions. Find the distance traveled by a point on the edge of the pulley. Solution: Since the pulley has a circumference of 6π and it makes.5 revolutions, the distance traveled by a point on the edge would be.5(6π) inches. Also, the pulley turned through 540 since 540 = (.5)360. Example 7: Suppose a wheel spins at 800 rpm (revolutions per minute). If a nail on that wheel is 4 inches from the center, how far does that nail travel in one minute? What is its speed in miles per hour? Solution: To answer this, note that in one revolution the nail travels once around the full circle of radius 4 inches. That is a distance of C = π inches. In one minute that wheel turns 800 revolutions, so the nail travels a total of (800) ( ) 7037 inches in one minute. This large number is converted to feet and to miles by the standard methods: (7037) () feet, which equals about (5864.3) (580). miles.

10 Section 6. Area and Arclength 435 Therefore, the nail travels. miles in a minute. If it continues going for one hour at that speed it will go 60 times as far, that is it will travel (60) (.) 66.6 miles. The nail is going more than 66 miles per hour. Exercises 6. Use the following information to find the arclength and area of the sector in a circle with central angle θ. Your answer should be accurate to three decimal places.. θ = 45, radius of length feet. θ = 37, radius of length 0 cm 3. θ = 30, radius of length.5 meters Find the central angle θ. Round your answer to the nearest degree. 4. The area of the circle is 34.5 square meters and the area of the sector formed by θ is square meters. 5. The circumference of the circle is feet and the arclength subtended by θ is feet. 6. The radius of the circle is 6 miles and the arclength subtended by θ is 6 miles. 7. The diameter of the circle is 45 cm and the area of the sector is sq cm. Solve the following problems. 8. The area of a sector is 5 square inches and the central angle that subtends this sector measures 6. Find the radius of the circle and the arclength that the angle subtends. 9. There are two pizzas, a 0-inch pizza and a -inch pizza. The 0-inch pizza is cut into six pieces and the -inch pizza is cut into eight pieces. For each pizza the slices are the same shape and size. Which pizza has the most food per slice? 0. An LP record travels at 33 3 rpm (revolutions per minute). If the record has a diameter of inches, how fast is a point on the edge of the record traveling (in inches per minute)? How fast is a point that is half-way between the center and the edge traveling (in inches per minute)?. The second hand on a clock is 6 inches long. How far does the tip of the hand move in 40 seconds?. What is the speed (in miles/hour) of a point on the tip of the second hand in question? 3. A neutron star (a remnant of a supernova explosion) is approximately the size of the earth but spins at a rate of revolutions per second. How fast (in miles/second) is a point on its equator moving? (Assume that the radius is 4000 miles.) 4. A bicycle that has wheels with a 0-inch diameter is traveling at 35 miles per hour. How fast are the wheels turning in rpm (revolutions per minute)?

11 436 Chapter 6 Ratios 6.3 Geometry Review: Similar Triangles Similarity is defined in the dictionary as having characteristics in common. We often use this definition of similarity when comparing two things. We say that two cars are similar when they have some of the same features. Likewise, people often notice that a parent and child are similar in their actions. When we say this are we saying that they have the same temperaments or that their body language is the same? This is not a very exact way to compare two things since we aren t sure which characteristics the two things have in common. In Geometry, however, we use a more precise definition of that word: two objects are similar when they have the exact same shape, but not necessarily the same size. In our everyday lives, we often encounter objects that are similar to an original object, such as a map of the county, a reduced photocopy, or a scale model of a train. Changing the size of an object without changing the shape gives us an object that is similar to the original. Two mathematically similar objects will also satisfy the dictionary definition because they will have many characteristics in common. It is those properties shared by similar objects that are important in Mathematics. Let s investigate some of the characteristics that two mathematically similar geometric figures will have. Figure shows two similar geometric figures. These figures happen to be pentagons. Note that they have the same shape but different sizes. A A D D B C F E B C E F Figure If you were to measure all of the interior angles in both objects, you would find that each interior angle has the same measure as its counterpart in the other figure. For example BAD = B A D. All geometrically similar figures will have the same

12 Section 6.3 Geometry Review 437 interior angles. This is because the interior angles in a figure actually determine the shape of the figure. Similar geometric figures have another useful characteristic. By measuring distances in the picture above we can see that the segment A B is twice as long as the segment AB and B C is twice as long as BC. These relationships are written in symbols as: A B = AB and B C = BC. Here we have a scale factor of. Take any two points X, Y in the first figure and suppose X, Y are the corresponding points in the second figure. Then having a scale factor of means that X Y = XY. Measuring other distances helps verify this observation. For instance, compare CD and C D. This relationship between distances justifies the statement that the larger figure has the same shape as the smaller figure but is exactly times as large. Different scale factors can be used to produce similar figures of different sizes. Here are two more examples. A D A D C B E B F C F E Figure What are the scale factors of these two pentagons, compared to the original one F in Figure? We can determine them by computing ratios. If z is the factor needed to transform figure F to figure F then A D = z AD and D E = z DE, etc. A D D E Therefore the scale factor z is the ratio z = =. In this example, z = AD DE / and we see that F is similar to F but / as large. What is the factor z needed to convert figure F to the figure F pictured above? The similarity of the figures gives us that the ratios of corresponding sides (and lengths) are all the same.

13 438 Chapter 6 Ratios For this course, triangles are the most important geometric figures since they allow us to solve many real-world applications. But before we investigate similar triangles, let us review some of the other properties of triangles. A triangle has three corners (or vertices), three sides, and three interior angles. The standard way to name these quantities is to use capital letters for the corners, and to use the same letters for the angles. We use the corresponding lowercase letters to label the sides opposite those corners. The labels in Figure 3 illustrate this notation. B c a A b Figure 3 C Note that the letter A stands for the corner point as well as for the angle BAC. One important property of triangles is that the sum of the interior angles is always equal to 80. Why is this true? For the curious student, a proof of this is given as an exercise. Another property is that the longest side is always opposite the largest angle and the shortest side is always opposite the smallest angle. These are useful facts to remember when checking the validity of a solution. Often you need to compare two triangles in order to solve a problem. Two triangles that are exactly the same size and shape are called congruent. Figure 4 shows two congruent triangles. Congruent triangles are similar and have the same size. Figure 4 From geometry class you may recall that there are several different ways to determine whether two triangles are congruent. One of these is the side-side-side (SSS) theorem which states that if each side of a triangle can be matched with a side of equal length in another triangle, then the triangles are congruent. Other methods are the side-angle-side (SAS) and angle-side-angle (ASA) theorems.

14 Section 6.3 Geometry Review 439 The last property of triangles that we discuss here is the Pythagorean Theorem. This property is true only for right triangles (i.e. for triangles with a 90 angle). This theorem states that the square of the side opposite the right angle is equal to the sum of the squares of the other two sides. The side opposite the right angle is called the hypotenuse, usually labeled c, and the other sides are called the legs, usually labeled a and b. With this labeling, the Pythagorean Theorem states that a + b = c. This fact can be proved using similar triangles and is given later as an exercise for curious students. Let s now look at an example where we explore two similar triangles. Figure 5 illustrates the general case for similar triangles. Y Y X z y x z Z x X Z y Figure 5 Suppose the triangles X Y Z is similar to triangle XYZ. We will repeat the remarks about ratios of sides in this context. From similarity, there is a scale factor k so that X Y Z is exactly k times as large as XYZ. Therefore the side lengths satisfy the equations: z = k z y = k y x = k x From this, we can solve for k to find: z y x = k = k = k z y x This says that the ratios of corresponding sides are equal and we have the following: z y x = = z y x This is an important property! It is often used to find the sides of similar triangles, and it is crucial for the description of trigonometric functions in Chapter 9. To apply these powerful tools concerning similarity to triangles, we must first establish some rules for determining when two triangles are similar. Just as with congruent triangles, there are several ways to do this.

15 440 Chapter 6 Ratios One way is to use the definition: compare corresponding angles. This is called the angle-angle-angle (AAA) theorem for similarity. In fact, since the sum of the angles in a triangle must equal 80, it is enough for us to know that just two of the corresponding angles have equal measures. This is called the angle-angle (AA) theorem for similarity. The theorems for determining similar triangles are summarized in the list below. Properties of Similar Triangles Two triangles are similar if and only if any one of the following statements is true.. (SSS) The ratios of all three pairs of sides are equal.. (AA) Two pairs of angles have the same measure. 3. (SAS) The ratios of at least two pairs of sides are equal and their included angles have the same measure. These properties allow us to determine when two triangles are similar. Because of this, we can use the characteristics of similar triangles to solve the following problem, and others like it. A student on the Oval wants to estimate the height of the Main Library. At 3 pm, the student notices that he casts a shadow that is 4.5 feet long. He also measures the length of the shadow cast by the building to be 00 feet. The student knows that he is 6 feet tall. How can he use this information to estimate the height of the building? The problem is pictured below using triangles (not drawn to scale). Since the angle of elevation for the sun is the same for both shadows and we assume that both the student and the building form right angles with the ground, the two triangles are similar by the AA theorem. Therefore we can use the ratios of the corresponding sides to solve the problem. 6 feet x feet The equal ratios are: 4.5 feet 00 feet x 6 = Solve for x to find that x = feet. [Comment: This may not be an accurate estimate for the height of the Main Library.]

16 Section 6.3 Geometry Review 44 Example : Given the following information about triangles ABC and A B C, show that they are similar, and then solve the triangles (i.e. find all the angles and sides). I. A = 49, B = 30.53, a = 5., b = 3.5, c = 6.78 C = 00.47, a = 3., b =. Solution: Since we know A and B, we can compute C = = To show that the triangles are similar, we must check that the ratios of the sides are equal. a 3. b. = =. 6 and = =. 6 a 5. b 3.5 a b Since C = C and =, the triangles are similar by the SAS theorem. a b So the corresponding angles are equal A = A = 49, B = B = Triangle A B C is.6 times as large as ABC so c =.6 c =.6(6.78) = II. C = 90, B = 37, a = 6, b = 4.5 C = 90, A = 53, b = 9 Solution: ABC is a right triangle with hypotenuse c. We can use the Pythagorean Theorem to find c, accurate to two decimal places. ( 6) + ( 4.5) = c 7.5 = c = c Angle A can be found by taking so A = 53. Triangle A B C is similar to ABC by the AA theorem since A = A and C = C. So B = 37. Similarity says that the ratios of the corresponding sides are equal, and we use that to find a and c accurate to two decimal places. a 9 = a = 6 = and c 9 = c = 7.5 =

17 44 Chapter 6 Ratios Example : A mirror can be used to measure the height of an object. A small mirror is placed on level ground 4 feet away from a flagpole. A person who is 66 inches tall sees the top of the pole in the mirror when she is feet away from the mirror. What is the height of the flagpole? Solution: The situation is represented by the picture below. We must covert all of the measurements into the same units so we will covert everything into inches. feet is 4 inches and 4 feet is 48 inches x inches 66 inches 48 inches 4 inches In order to solve the problem we need to make sure that the two triangles are similar. This can be shown using the AA theorem. We assume that the person and the flagpole form right angles with the ground. We also know that the two angles that are formed at the mirror are congruent (why?). This gives us that the two triangles are similar and therefore the ratios of the corresponding sides are equal. Finding the value for x can be done by inspection. Since 48 is twice as big as 4, x must be twice as large as 66, so x = 3. The flagpole is 3 inches or feet high. [Comments: What aspects of this problem are not truly accurate? For example: Her eyes are not 66 inches above the ground. The pole might not be straight, the mirror is not a single point, etc.]

18 Section 6.3 Geometry Review 443 Exercises 6.3 Find the missing angles and sides. (Round answers to decimal places.) First, state whether the two triangles are similar and how you know. Then, if possible, find the missing pieces of the given triangle(s) using only the information given and the properties of similar triangles. (Round answers to decimal places.) Note: The drawings are not done to scale

19 444 Chapter 6 Ratios If possible, solve the following triangles ABC and A B C using only the properties of similar triangles. (Round answers to decimal places.) 7. a = 3.5, b = 6, C is a right angle. a = 7, c = 3.9, C is a right angle and A = a = 5, b = 0, B = 5 a = 0, b = 0, B = 5 9. a = 7, b = 4, c = 4.8, A = 05, B = 33.5 a = 3.5, c =, A = 05, C = 33.5 (Be careful!) 0. a = 5, b = 5, C is a right angle. a = 7.5, b = 7.5, C is a right angle. Solve the following application problems to decimal place accuracy.. A scale model of a pyramid is similar to the original and is 5% the size. If the model is 4 feet tall, how tall is the original pyramid?. A scale model of a ship is similar to the original. The scale model is 8 inches long and the mast is 4.5 inches high. If the real ship is 65 feet long, how high is the real mast? 3. Suppose a 6-foot tall person casts a 0-foot shadow. How long is the shadow cast by the 8.3-foot tree that is next to the person? 4. You are at Evel Knievel s final stunt performance. You want to know the height and length of the ramp that he is going to use. For safety reasons, you are not able to get close to the ramp. height length Labeled drawing of the ramp. (Not drawn to scale) 4 feet

20 Section 6.3 Geometry Review 445 (a) Someone tells you that a smaller triangle with the same angles as the ramp measures 6 by.4 by Use this information to find the height and length. (b) Assuming that you can get next to the ramp, can you think of an easy way to find the measurements of a similar triangle, and therefore the ramp itself? 5. You are on the bank of a river and want to find the distance across the river. You have no way to cross the river so you will use similar triangles to find the distance across. To do this, you find two trees that are on opposite sides of the river, directly across from each other. You then measure a distance of 0 feet downstream from the tree on your side of the river. Call this point A. From point A, you then measure the angle between the trees to be 67. You measure the angle located at the tree and find that it measures 95. To find the distance between the trees (and thus the distance across the river), you must use a piece of paper, a protractor (a tool used to measure/draw angles), and a ruler. Use these tools and similar triangles to find the approximate distance across the river. (Hint: Draw a similar triangle on the paper.) Food for Thought 6. Where have you encountered similar figures in your everyday life? 7. A baby is a small version of an adult. Are the body shapes of an infant and an adult similar figures? Extra Exercises 8. Prove that the sum of the angles in a triangle equal 80. (Hint: Look at the pattern grid made from copies of a given triangle.) 9. Prove the Pythagorean Theorem by using similar triangles. (Hint: Use the picture below and find the three similar right triangles contained in the picture.) C b a A c x F x B

21 446 Chapter 6 Ratios Chapter 6 Review Variation For problems 4, find a formula that illustrates the relationship between the two quantities. This involves finding a constant value k.. A varies directly as the cube of r. When r = 9, A = 97.. K varies inversely as L. When K = 7, L = J varies jointly as M and N. When M = 30 and N = 9, J = The relationship between grams and kilograms is a type of variation. What type is it? For problems 5 7, solve. 5. Because it is often more cost efficient to produce items in large numbers, the cost of producing CD s is inversely proportional to the number produced. If it costs $00 to produce 4000 CD s, how much does it cost to produce 3500 CD s? How much does it cost to produce 8000 CD s? 6. Over a fixed distance, the time it takes to travel the distance varies inversely to the speed. If it takes 30 seconds to travel the distance at 45 mph, find the time it takes for a car traveling 50 mph to travel the distance. 7. The price paid to fill the gas tank of your car varies directly with number of gallons purchased. If you pay $40 for 5 gallons of gas, how many gallons can you purchase for $50? Circular Sectors For problems 8 0, find the missing information for each circle. 8. Given θ = 45 and the radius is 7.5 cm, find the area of the sector and the arclength. 9. Given θ = 0 and the diameter is in, find the area of the sector and the arclength. 0. Given that the area of the sector is 6.35 sq. ft. and that area of the circle is sq. ft., find θ, the circumference, and the arclength. For problems &, solve.. A wheel on a motorcycle has a diameter of 4 inches. How many revolutions per minute is the wheel making if the motorcycle is traveling at 60 mph?. The hour hand of a clock is 7 inches long. How far does the tip of the hour hand move in 45 minutes? Similar Triangles For problems 3 5, when possible, solve the triangles ABC and A B C. Use only the properties covered in section 6.3. (Round answers to decimal places.) 3. a = 3.3, b = 7.54, c = 7.69, C = 80 : a = 8.5, b = 8.85, B = 75, C = A = 90, B = 5, a = 5, b =.: A = 90, C = 65, b = a = 0.35, b =.4, c = 5.7: a = 7.45, b = 77.98, c = 3.6

22 Chapter 6 Section 6., page 47. It will double. 3. Direct variation: s = 60m or m = s A 9. 5 y = x V = πr = π 3 hr. Approximately 35 mph grams 5. Approximately 3.67 ohms Section 6., page ft,.57 sq ft ft, sq ft 5. θ = θ = inch pizza with 4.4 sq in. 8π 5.33 inches Answers to Odd-Numbered Exercises A Similar by SAS. c = 6.95, b = A = 30.6, B = B = Similar by AA b =.4 B = C = feet feet 5. Approximately 9.8 feet Chapter 6 Review, page A = r 3 3. J = MN 5. $8.57, $ gallons 9. Area S = 0.56 sq. in. Arclength =.9 in revolutions 3. Similar by SAS: c = 9.5 B = 75, A = A = 5 5. Not similar. 3. Approximately mi/sec Section 6.3, page 443. side: 30 angle: Not similar 5. Similar by SSS. other angle: 4.54