Optimization. y x + y 40 x 16 y 20 x : the number of hours spent cleaning the premises y : the number of hours spent working in the kitchen

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1 Optimization 1 Jonathan works at a golf club working his summer vacation. He sometimes cleans the premises and sometimes works in the kitchen at the club's restaurant. Jonathan makes $8 per hour when cleaning the premises and $9.50 per hour when working in the kitchen. There are certain constraints on the number of hours he can devote to each job every week. This situation is represented by the system of inequalities and the polygon of constraints given below. y 0 y 0 + y y 0 : the number of hours spent cleaning the premises y : the number of hours spent working in the kitchen Q R Coordinates of the vertices of the polygon P(16, 0) P S Q(16, 0) R(0, 0) S(40, 0) This week, Jonathan s employer informed him that there would be an additional constraint. This new constraint is represented by the following inequality: y + 0. With this new constraint, by how much will Jonathan's maimum possible income decrease?

2 To raise money, the Graduation Committee decides to sell cases of fruit. The following polygon represents the constraints that must be respected. If represents the number of cases of oranges for sale and y, the number of cases of grapefruit for sale, the constraints are: y C y + y y D B A For each case of oranges and grapefruit sold, the Graduation Committee makes a profit of $1.00 and $1.50, respectively. Yesterday, the head of the committee received a call from the supplier. Because of a recent flood, the supplier can deliver a maimum of 400 cases of fruit. By how much will the maimum possible revenue decrease because of the flood? 3 Si inequalities are represented by the polygon of constraints below. l 1 0 y l y 0 l 3? l 4 + y l 3 l 4 l 5 + y 0 l 6 y l 3 forms a right angle with l l 6 l Which of the following is the missing in equality (l 3)? A) + y 30 C) y 30 B) + y 30 D) y 30

3 4 The city council of a town wants to minimize the cost of staffing its recreation centres in the summer months. The council has determined that supervisors will be paid $3500 for the summer and staff workers will be paid $1500 for the summer. The council wants to hire its employees using the following constraints: The maimum number of employees for its centres is 30 and the minimum is 18. The council also wants to hire at least 6 supervisors but no more than 14 supervisors. It wants to hire at least 8 staff workers. The number of staff workers must be at most twice the number of supervisors. How many staff workers and how many supervisors can the town council hire and minimize its costs? 5 Wheeler is a producer of mountain bikes and road bikes. Because of its small size, it can build no more than 80 bikes each week. To meet certain conditions in its workshop, it must build at least 45 mountain bikes, and at least 10 road bikes weekly. To meet consumer demand, it must manufacture at least 3 times as many mountain bikes as road bikes. The following is the system of constraints for Wheeler's weekly bike production: = the number of road bikes produced weekly y = the number of mountain bikes produced weekly 0 y 0 10 y 45 + y 80 y 3 For each road bike and mountain bike produced, Wheeler earns a profit of $50 and $175, respectively. What is the maimum weekly profit that can be earned? 6 A fisherman has to separate his daily catch of shellfish into two categories before he can sell them. Lobsters are sold for $8.70 each and crabs are sold for $9.60 each. On an average day, the fisherman can epect to catch a minimum of 35 crabs and a maimum of 60. By eperience, there are at most twice as many lobsters as crabs in a daily catch and never has the fisherman caught more than 140 shellfish in a single day. Using a polygon of constraints, determine the maimum revenue that this fisherman can epect to make

4 7 Instruments Quebecois makes two types of graphing calculators, the Gold Edition and the Bronze Edition. In order to meet daily demands, it must make at least 00 Gold Editions and at least 100 Bronze Editions. The factory produces at least twice as many Gold Editions as Bronze Editions, but can make no more than 600 calculators a day. Given : number of Gold Editions y: number of Bronze Editions Constraints: 0 y 0 00 y 100 y + y 600 The profit on a Gold Edition is $0.00 and $15.00 on a Bronze Edition. In the answer booklet, graph the polygon of constraints and determine the number of calculators that the company should make to maimize its profit. 8 The Grad Committee plans to sell chocolate bars to raise money for its upcoming dance. This year the committee members have decided to sell two types, one with roasted almonds and the other with caramel. They have a maimum of 500 bars to sell. They epect to sell a minimum of 10 almond chocolate bars. From past eperience, almond chocolate bars sell at most 4 times as well as caramel ones. They make a profit of $0.80 for each almond chocolate bar and $1 for each caramel chocolate bar. Let : number of almond chocolate bars y: number of caramel chocolate bars What is the difference in the maimum profit if they had epected to sell a minimum of 160 almond chocolate bars rather than 10?

5 9 Murray plans a trip to New York in July. In order to save money, he works at two different part-time jobs on weekends. At the first job, he works a minimum of 10 hours per month and at the second, a maimum of 40 hours per month. Murray must work at least 30 hours per month but no more than 60 hours per month. He must work at least as many hours at the second job as he does at the first. He makes $6.30 an hour at the first job and $8 an hour at the second job. Let : number of hours per month at first job y: number of hours per month at second job The initial constraints for this situation are: 10 y 40 y 0 y 30 y 60 y Because of a shortage of employees, Murray was later advised that he could increase the number of hours he worked at the second job. By how much did Murray s maimum possible salary increase because of the employee shortage?. 10 Kim is organizing a fundraiser for her soccer team. She will sell hot dogs and hamburgers outside a popular grocery store. She needs to purchase enough supplies to be able to make the following: at most 800 hot dogs and hamburgers at least 150 hot dogs a minimum of 100 but not more than 400 hamburgers at most twice as many hamburgers as hot dogs Her cost is $0.45 per hot dog and $0.75 per hamburger. She will sell the hot dogs at $1 each and hamburgers at $1.50 each. Let : the number of hot dogs y: the number of hamburgers Given her constraints, how many hot dogs and hamburgers does Kim need to sell to make the greatest profit possible?

6 Answers 1 With this new constraint, Jonathan's maimum possible income will decrease by $15. The decrease in revenue caused by the flood is $15. 3 D 4 The town should hire 6 supervisors and 1 staff workers in order to minimize its costs. 5 The maimum weekly profit is $ The maimum revenue this fisherman can epect to make is $17. 7 Instruments Quebecois should produce 500 Gold Editions and 100 Bronze Editions to maimize profit. 8 The difference in the maimum profit is $8. 9 Murray s maimum possible salary increased by $17 10 Kim needs to sell 400 hot dogs and 400 hamburgers to make the greatest profit.

7 Trigonometry The population growth P of the native people of Northern Quebec is represented by the graph of the sinusoidal function with equation P 4 t 400 sin t where t is the number of years elapsed since Which one of the graphs below corresponds to this situation? A) C) E(t) E(t) t t B) D) E(t) E(t) t t 3 Prove that, sin tan 1 1 cos tan sec 1 cos

8 4 Peter and Lucy are studying the repeated up and down movement of a piston in a gasoline engine. Peter claims that the piston moves according to the rule : d sin t cos t sin t cos t sin t cos t Lucy believes that the piston moves according to the rule : d = sec t Prove that Peter and Lucy are both correct by showing that sin t cos t sin t cos t sin t cos t sec t 5 A sinusoidal function is shown below. f() Which of the following rules define this function? f 4sin 6 A) 6 B) f 4sin f 4sin 6 C) 6 D) f 4sin

9 6 f -100 sin 4 4 Given the function 100 Which of the following is true? A) 6 is a zero of the function f on the interval [0, 8]. B) The function f is decreasing on the interval [4, 8]. C) Dom f = and range f = [-100, 100]. D) The period of the function f is 8 and the phase shift is Prove the following trigonometric identity: 1 cosec A (cosec A + cot A) = 1 cos A 8 For all values of A (for which A is defined), the epression tan A + cot A is equal to A) sin A cos A. C) sec A cosec A. B) sec A cos A. D) sin A cosec A. 9 tan A The epression sec A is equivalent to cosec A A) cos A cos A C) sin A B) cos A D) (1 sin A) sin A cos A

10 10 t The equation of a sinusoidal function is f(t) = sin + 4 where t [0, 4[. On which interval is this function increasing? A) [0, ] [3, 4[ B),, C),,,, D) [, 6] 11 sin t sec t Prove the following identity : sec t = -1. cot t 1 Many factors influence the deer population in a given habitat: climate, hunting, predators, etc. The following graph shows the evolution of a population of deer as a function of time. Number of deer f() Write the rule that can be used to represent this function Time (years)

11 13 What is the domain of the tangent function? A) { = (k + 1) where k Z} B) { (k + 1) where k Z} C) { k where k Z} D) 14 In which of the following intervals does the sine function decrease? - A), 3 C), 5 B), 6 6 D) ], [ 15 The amplitude of function f() = a sin ALWAYS has a value of... A) C) a B) D) a 16 Given the trigonometric equation: cos + cos = 1, [, ] What are the eact values of that satisfy this equation?

12 17 The following diagram (not drawn to scale) represents a predator-prey situation of the population of rabbits and foes in a region of Mt.Tremblant Park, both of which follow a sinusoidal model. Population 1000 rabbits foes 4 Years after 1990 Consider t = 0 to be Initially there are 1000 rabbits and 50 foes. The graph shows the rabbit population increasing for 4 years then decreasing to its minimum population of 800 in 00. Over the same period of time the fo population starting from its maimum decreases, reaches its minimum, then increases to a population of 175 in 00. If the model continues as shown, what is the difference between the rabbit and fo populations in 004? 18 A hyperbola and a trigonometric function are drawn on the same Cartesian plane. The equation of the hyperbola is y y The foci of the hyperbola are directly below two of the maima of the trigonometric function. Which of the following is an equation of the trigonometric function? F F 1 A) y 8.8 cos C) y 3.3 cos B) y 8.8 cos D) y 3.3 cos 11 11

13 19 A cuckoo clock uses a pendulum to keep time. The movement of the pendulum can be described by a sinusoidal function. The length of the pendulum is 31 cm. At its lowest point, the pendulum is 1.5 m from the ground. t 31 cm t 3 t 1 The pendulum starts its movement at t 1. The interior angle between t and t 3 is 90 and it takes the pendulum second to go from t to t m What is the height of the pendulum relative to the ground after 1 hour? Round your answer to the nearest centimetre.

14 0 Tom s ride on the Ferris wheel at La Ronde can be described by the graph of the sinusoidal function shown below. The graph represents the height of Tom s seat above the ground, in metres, as a function of the time, t, in seconds. The distance between the minimum and maimum heights of Tom s seat is 0 metres. Tom s seat reaches its first maimum height 15 seconds after the Ferris wheel begins to turn. H(t) 15 t How many metres above ground is Tom s seat 0 seconds after the Ferris wheel begins to turn?(assume Tom s seat started from rest at the bottom of the Ferris wheel.) 1 Given cos + 3 sin 3 = 0, 0,. What are the eact values of? In the sinusoidal function below, represents time in seconds. f 5 sin 1 3 Which of the following correctly describes the properties associated with the sinusoidal function? A) Maimum Value: 5 Minimum Value: Period: sec C) Maimum Value: 8 Minimum Value: - Period: 4 sec B) Maimum Value: 8 Minimum Value: 3 Period: 5 sec D) Maimum Value: 5 Minimum Value: 3 Period: ( ) sec

15 3 The depth of water at the port of St. Marie-Elise varies according to the tides. A sinusoidal function can be used to predict water depth. At low tide, the depth of the water is 9. m. At high tide, it is 18. m. The time between two low tides is 1 hours and 30 minutes. In order for an oil tanker to dock safely, the depth of water must be at least 14.5 m. For how many hours can an oil tanker dock safely between two consecutive low tides? 4 A fountain in a shopping centre has a single jet of water. The height of the jet of water varies according to a sinusoidal function. Joel notes that, in eactly one minute, the jet goes from a minimum height of 1 m to a maimum height of 5 m and back to 1 m. At 13:00, the jet of water is at a height of 1 m. What will be the height of the jet of water, to the nearest tenth of a metre, when the clock reads 13:1:40? (13 hours, 1 minutes, 40 seconds) 5 The average temperature of a fictitious town is given by the equation () = 0 sin 13 + where represents the number of weeks since the beginning of the year. During the course of the year, for how many weeks was the average temperature higher than 15 C?. 6 During an eperiment, the intensity i(t) of the electric current of a device as a function of time t elapsed since the beginning of the eperiment is given by: i(t) = 6 sin t where t is epressed in seconds. The device emits a sound signal each time the current s intensity is equal to 9. The eperiment lasts 10 seconds. How many sound signals does the device emit during the eperiment?

16 ANSWERS A 5 A 6 A 8 C 9 B 10 A 1 f sin 4 Answer 500 f sin 4 4 or 500 f cos 4 or B 14 C 15 D 16 The eact values of are -, -,, 3 3

17 17 The difference between the rabbit and the fo populations is A 19 To the nearest cm, the pendulum will be 156 cm above the ground after 1 hour. 0 Tom s seat is 17 m above the ground 0 seconds after the Ferris wheel begins to turn. 1 The eact values of are 6 or. C 3 The oil tanker has 5.5 hours to dock safely between two consecutive low tides. 4 At 13 hours 1 minutes and 40 seconds, the water jet will be at a height of 4 m. 5 During the course of the year, the average temperature is higher than 15 C for 14.8 weeks. 6 During the 10 seconds, the apparatus emitted 10 sound signals

18 Sinusoidal Functions 1 The tuning fork is a device used to verify the standard pitch of musical instruments. The international standard pitch has been set at a frequency of 440 cycles/second. Write a rule in the form f(t) = A sin Bt that epresses this oscillation where t represents the number of seconds. 13 An oscillatory movement is epressed by the equation f(t) = 160 sin(500t 100). Find a) its frequency b) its period c) its phase shift d) its amplitude 14 A sound wave is described by a sinusoidal movement whose equation is : f(t) = -30 sin(30t 0). Find a) the amplitude b) the period c) the frequency d) the phase shift 15 The pendulum of a Grandfather clock completes 0 cycles/minute. It moves a distance of 30 cm. Write a rule in the form f(t) = A sin Bt which epresses the movement of the pendulum where t represents the number of minutes. 30

19 16 Radio station CKOI-FM broadcasts through a frequency of 97 KHz, or cycles/s. The radio's volume is set at 5, thus determining the sound amplitude. Write a rule in the form f(t) = A sin Bt of the sinusoidal curve representing the sound waves transmitted. 17 For each of the following graphs, find 1) the amplitude ) the period 3) the frequency 4) the phase shift 5) the equation a) b) The number of pairs of shoes manufactured by a factory from December to June is associated with the sinusoidal function illustrated below. where t is the number of months elapsed since December and n(t), the number of pairs of shoes. What is the rule of function n?

20 19 Jan is doing research on the phenomena of vibrations. She compiles a series of results and obtains the following graph on the computer screen. f() What rule represents this function? A team studies variations of certain physical phenomena. As you can see from the diagram on the right, for three seconds, the graph appearing on the oscilloscope is that of a sinusoidal function. f() 7 Write the rule of correspondence for this sinusoidal function. 3-7

21 1 The given clock only has a second hand. At noon, Maria sees the tip of the second hand at the top of the clock and notes the height of the tip of second hand in relation to the bottom of the clock. The second hand is 10 cm long. What is the function rule describing the height (h) in cm of the tip of the second hand compared to the time (t) elapsed in seconds since noon? Function f is defined by the rule f() = -sin. Determine the zeros of this function over [0, 4π[. 3 The screen of the oscilloscope below illustrates a sinusoidal function f, representing the amplitude of the vibration of a guitar string as a function of time t, in seconds. f ( t ) (5, 7) (0, 4) (15, 1) t What is the rule of this sinusoidal function?

22 4 For two years, oceanographers have compiled data on the mass of humpback whales. They noted that the whale's mass varies according to the following sinusoidal function: m(t) = 5 sin t where t [0, 4[ where t is the number of months gone by since the beginning of the study, and m(t) represents the mass of the whales in tonnes. When did the mass of the humpback whales reach eactly 100 tonnes during the period covered by the study? 5 The depth of water at the port of St. Marie-Elise varies according to the tides. A sinusoidal function can be used to predict water depth. At low tide, the depth of the water is 9. m. At high tide, it is 18. m. The time between two low tides is 1 hours and 30 minutes. In order for an oil tanker to dock safely, the depth of water must be at least 14.5 m. For how many hours can an oil tanker dock safely between two consecutive low tides? Correction key 1 f(t) = A sin 880 t or f(t) = A sin 764.6t 13 a) 50 b) 1 50 = c) 1 = d) a) 30 b) = c) 15 d) = The rule is f(t) = 15 sin 40t or f(t) = 15 sin 15.66t. 16 The rule is f(t) = 5 sin t or f(t) = 5 sin t.

23 a) 1) amplitude = 3 ) period = 8 3) frequency = 1 8 4) phase shift = 5 or phase shift = 1 5) f() = 3sin 4 b) 1) amplitude = ) period = 1 3) frequency = 4) phase shift = 0 5) f() = sin 4 ( 5) or f() = -3sin 4 ( 1) The rule that corresponds to function n is n(t) = sin The rule representing this function is f() = sin + 1 t The rule of correspondence for this sinusoidal function is : f() = 7 sin π 1 The rule describing the height (h) of the needle at time (t) elapsed since noon is: h(t) = 10 sin 30 (t 45) + 10 or h(t) = 10 cos 30 (t) The zeros of this function over [0, 4π[ are 5π 4, 7π 4 t The rule of the sinusoidal function is f(t) = 3 sin , 13π and 4 15π 4 4 During the period covered by the study, the mass of the humpback whales was eactly 100 tonnes at 3.54 months and 8.46 months. 5 The oil tanker has 5.5 hours to dock safely between two consecutive low tides.

24 Conics 1 Find the equation of the parabola which has its verte at the origin and its focus at point F in the following cases. a) F(, 0) b) F(0,-4) c) F(-3,0) d) F(0, 5) In the Cartesian plane, represent the regions defined by the following relation. 4y 3 Standing at the top of a cliff 3 m high, a person throws a stone which hits the ground 8 m from the base of the cliff. The following graph shows the trajectory of the stone. It is a part of a parabola whose verte is at the origin. What are the coordinates of the focus of this parabola? 4 The captain of a warship notices on his radar that a torpedo is approaching from the right. The head of the torpedo is the shape of a parabola. A cross section of the torpedo is shown in the Cartesian plane below. y Find the length (to the nearest centimetre) of the torpedo 50 cm from the tip of the head if the focus is 10 cm from the verte (tip).

25 5 A bridge is supported by a parabolic steel arch as shown in the following diagram. S h A F h B Distance from the focus to the verte : Height : 8 m 1 m Find the distance AB, which separates the feet of the arch. 6 The lampshade of a lamp is 30 cm in height. The top opening has an elliptical shape defined by the relation y The bottom opening has an elliptical shape defined by the relation y LAMPSHADE L Seen in perspective Based on the major aes of the ellipses What is the length of the L, the slant height of the lampshade? 7 Find the standard form of the equation of the ellipse given its foci F 1 and F and two vertices. F 1(-3, 0), F (3, 0) and V 1(-5, 0), V (5, 0)

26 8 Find the standard form of the equation of the ellipse given its foci F 1 and F and two vertices. F 1(0, -6), F (0, 6) and V 1(0, -7), V (0, 7) 9 A flower garden in the shape of an ellipse is laid out by placing two posts 6 m apart and attaching the ends of a 10 m cord to each post. What is the equation of the ellipse representing this flower bed if it is centered at the origin in the Cartesian plane? 10 A factory manufactures windows that are in the shape of a semi-ellipse. The most popular model correspond to the equation y A metal strip is glued all along the top and the bottom of this window as shown in the diagram below. A good approimation of the perimeter of an ellipse is found by using the formula [ 3(a + b) (a + 3b)(3a + b) ] where a and b represent respectively the length of each semi-ais in metres. Rounded to the hundredth of a metre, what is the sum of the lengths of the two metal strips? A) 4.00 m C) 4.60 m B) 4.14 m D) 5.67 m

27 11 A railway tunnel goes through a mountain. A cross section shows that the tunnel is in the shape of the upper half of an ellipse whose equation i 9 + 4y = 14 where the unit of measure is the metre. The tunnel's lighting consists of lamps located at the upper focus of the ellipse. Rounded to the nearest hundredth, how far from the ground are the lights? A) 3.61 m C) 5.33 m B) 4.47 m D) 7.1 m 1 The support for a swing is made of metal tubing in the shape of a parabola. (0, 4) y (0, 3.85) The verte of the parabola is situated at point (0, 4) and the point at which the swing's chains are attached coincides with the focus whose coordinates are (0, 3.85). What is the equation of the parabola?

28 13 A sculpture in the garden of a contemporary art museum consists of a circle and a parabola, as shown below. y A O B // C D The verte of the parabola coincides with the centre O of the circle which has the equation + y 3 10y + 79 = 0 where the unit of measure is the metre. The parabola, whose ais of symmetry is vertical, is constructed such that angle AOB is a right angle. Rounded to the nearest hundredth, what distance is there between bases C and D of the sculpture? 14 Following the final event in the school's track and field meet, each member of the first place team will receive a trophy. A cross section of the trophy is illustrated below. D y (cm) C 8 cm E B A O (cm) The base of the trophy is the shape of a curve whose equation is + 9y 36 = 0 for y 0. On top of this curve is a circle whose equation is + y 8y + 1 = 0. The circle is topped by a section of a parabola. The opening at the top of the trophy is level with the focus of the parabola. This opening is 18 cm wide. What is the total height of the trophy?

29 15 The draft plan of the table Vivian designed is illustrated at the right. y The curvature of the table leg corresponds to that of a hyperbola whose equation is A y = 18 5 where the unit of measure is the centimetre. F The line segment AB, passes through the focus F, and represents the height of the table leg. B What is the height of the table leg? 16 In the adjacent diagram, the centre of the circle is located at the upper focus of the ellipse defined by : y 9 y 36 1 F The circle passes through the ends, A and B, of the minor ais of the ellipse. A 0 B What is the degree measure of arc AB, situated inside the ellipse?

30 17 A packaging company decorates its square boes according to the model shown below. The equation of the circle in the model has the form : ( h) + (y k) = r The equations of the parabolas are : 8 y + 16 = 0 and 8 + y + 8 = 0 If the cartesian unit of measure is 1 cm, calculate the length of the diagonal of this bo. 18 The equation of the hyperbola shown below is 4 y 1 = 0. y F F 1 Find the distance between foci F 1 and F.

31 19 A fire station is parabolic in shape. It is 4 metres wide at ground level and its maimum height is 9 metres. Fire engines enter the station through three identical rectangular doors, each 4 metres high. The front of the fire station is sketched below. y m What is the area of the three doors, in square metres? 0 A company specializing in weight-lifting equipment is advertising a new type of barbell. A cross-cut view of the barbell is sketched below. The radii of the two hemispherical ends measure 3 cm each. The centre part of the barbell is defined by a hyperbola. Foci F 1 and F lie on segments AD and CG respectively and the vertices are 4 cm apart. y M A B C Length F F 1 D E G P What is the length of segment (m MP)?

32 A car race takes place on an elliptical track. The foci of the ellipse are 500 m apart. Luke and Nathalie, each located at one of the foci of the ellipse, were looking at the same car at a certain point during the race. At that moment, each person's line of vision made an angle of 45 with the major ais. Rounded to the nearest tenth, what is the shortest distance between Luke and the track? 3 The line whose equation is 4 + 3y 43 = 0 is tangent to a circle with centre C(3, ). What is the equation of this circle? 4 A geometric loci is defined by the set of points Q, whose sum of the distances from two fied points M and N is constant. Which of the following equations represent this geometric loci? ( h) ( y k ) A) ( h) + (y k) = r C) + = 1 a b ( h) ( y k ) B) (y k) = 4c( h) D) = -1 a b 5 Given the following: 1) The set of all points equidistant from a fied point ) The set of all points in which the product of the distances from two fied points is constant 3) The set of all points such that each point in the set is equidistant from a line and a fied point Which of the statements above refer to a conic section? A) 1 only C) 1 and 3 only B) 1 and only D) and 3 only

33 6 The equations of two parabolas are given below: y = -1 ( 4) = 8(y 7) The directri of the first parabola intersects with the directri of the second. What are the coordinates of this point of intersection? 7 The logo of a graphics company is shown in the diagram on the right. The logo is a rectangle A B divided by a parabola. In designing the logo, the company used the equation to draw the parabola. y = 50 Focus The rectangle is labeled ABCD. Side AD is on the directri of the parabola and its focus is on side BC. What is the length of side BC of the rectangle? D C A) 5 units C) 71 units B) 50 units D) 100 units

34 8 A carpenter places two nails on a rectangular piece of plywood as illustrated in the diagram below. Each nail is placed 0 cm from each of the smaller edges. The smaller edges are 60 cm in length. The carpenter uses these two nails as focal points to draw the largest ellipse possible on the rectangular plane. 0 cm F 1 F. 0 cm 60 cm. Find the area of the rectangular piece of plywood. 9 A dome, in the shape of a semi-ellipse, protects a tennis court, as shown below. 8 m? 0 m 3 m The height of the dome at the centre is 8 m and its span is 0 m. Cameras must be fied to the roof of the dome at a horizontal distance of 3 meters from its edges. At what height are the cameras from the ground?

35 30 A student in Math 536 was designing a Halloween mask. He decided to draw an ellipse with equation y 1 as illustrated below A B C He then drew two eyes, represented by circles with radii of units. The centres of the circles are located directly above the foci of the ellipse. Net he drew a parabola which intersects two circles at points A and B respectively. Points A and B are on a line connecting the centres of the circles. The parabola is also tangent to a verte of the ellipse at point C The final step was to draw a nose represented by the dark circle in the diagram. The centre of the circle is the focus of the parabola. What are the coordinates of the centre of the dark circle? 31 Consider a parabola with verte (-1, 4) and focus (-4, 4). Which of the following is the equation of the parabola in standard form? A) (y 4) = 1( + 1) C) (y 4) = -1( + 1) B) ( + 1) = 1(y 4) D) ( + 1) = -1(y 4)

36 3 Daphne wants to install a rectangular in-ground pool in her backyard. She wants the pool, measured in metres, to be inscribed within an elliptical cement surface, as illustrated. According to the design, the domain of the ellipse would be [, 10] and the range [6, 18]. The two ends of the pool pass through the foci of the ellipse. What is the perimeter of Daphne's pool, to the nearest tenth of a metre? Answers: 1 a) y = 8 b) = -16y c) y = -1 d) = 0y 3 4 The coordinates of the focus are The width is 89 cm. 1 0, -. 5 The distance AB is 39. m. 6 L 30.4 cm

37 7 5 y y y C 11 B 1 The equation of the parabola is = -0.6(y 4) 13 Rounded to the nearest hundredth, the distance between C and D is 4.47 m. 14 The total height of the trophy is 8 cm. 15 The height of the table leg is 86.4 cm. 16 The measure of arc AB is 60 degrees. 17 The length of the diagonal is cm. 18 The distance between F 1 and F is 7.7 units. 19 The area of the three doors is 71.5 m.

38 0 The length of segment is 11 cm. The shortest distance between Luke and the track is metres. 3 The equation of the circle is ( 3) + (y ) = 5. Or + y 6 4y 1 = 0 4 C 5 C 6 The point of intersection is (3, 5). 7 B 8 Rounded to the nearest cm, the area of the plywood is 3900 cm. 9 The cameras are 5.71 m from the ground The coordinates of the centre of the dark circle are C 7 0, - 4 or (0, ). 8 3 To the nearest tenth of a metre, the perimeter of the pool is 8.6 m.