Solve the following system of equations using either substitution or elimination:

Transcription

1 Mathematics 04 Final Eam Review Topic : Solving a system of equations in three variables. 009 Solve the following system of equations using either substitution or elimination: + y + z 0 + y z y + z Select equations and eliminate one of the variables. nd and rd equations can be added to eliminate z and y. + y z y + z 4 Substitute this for in the st and nd or rd equations: + y + z 0 st equation y + z y + z nd equation y + z 4 Now solve the system of two equations in two variables: y + z multiply this equation by y + z 4 y + 4z 6 y + z 4 5z 0 z add the equations Substitute and z - in one of the original equations + y + 0 st equation to find y. + y 4 0 y y (,y,z) (,, -) 008 Algebraically solve the following system of equations using substitution or elimination: y z 5 6y z y z Solve the following system of equations using either substitution or elimination: + 5y + z 0 y + z y + z 8

2 Topic : Solve a system of equations in two variables using matrices. 009 Barry s Bakery sells cookies and muffins. One customer orders 5 muffins and 8 cookies, and is charged $7.50. Another customer orders muffins and 5 cookies, and is charged $0.75. Create a system of equations to represent this information and, using matrices, find the price of each item. Let price of muffins and y price of cookies 5 + 8y y A B Matri equation: A X B X A B y y y Lorraine buys 6 cheap golf balls and 4 epensive ones for $.50. Bob buys 4 cheap and epensive balls for $9.00. Create a system of equations to represent this information and, using matrices, find the price of the two kinds of balls. 007 The student council is having a dance as a fundraiser. They are selling two kinds of tickets: singles and couples. On the first day, 9 singles tickets and 4 couples tickets are sold for a total of $ On the second day, singles tickets and 8 couples tickets are sold for a total of $ Create a system of equations to represent this information and, using matrices, find the cost for each type of ticket.

3 Topic : Finding a quadratic equation using a system of equations. 009 Amy kicked a soccer ball. After the ball travelled a distance of 8 metres, it was at a height of. metres. When the ball hit the ground, it was 40 metres away from Amy. Set up and solve a system of equations to determine the quadratic function that models the path of the soccer ball. Use the three points to find an equation of the form y a + b + c (Find a, b, and c.) (,y) (0,0) 0 a 0 + b 0 + c c 0 c (8,.). a 8 + b a + 8b one equation (40,0) 0 a 40 + b a + 40b second equation Solve the system. 64a + 8b 0 600a + 40b You can use any method you like (ie. substitution, elimination, matrices) Elimination: Multiply first equation by -5 and add to second equation. 6 0a 40b 0 600a + 40b 6 80a 6 a b b 0 40b 0 b.5 a -.05, b.5, c 0 substitute these numbers in y a + b + c Equation: y or y A toy rocket is launched from a platform 5 metres above the ground and reaches a height of 9 m in 4 seconds. The rocket touches down on the ground after 0 seconds. Write an equation in the form h() t at bt c that describes the path of the rocket where t represents time in seconds and ht () represent the height of the rocket in metres. Height (4, 9) (0, 5) (0, 0) Tim e 007 An arrow is fired into the air from a height of metre. After seconds, the arrow is 9 metres in the air. After 4 seconds, the arrow is metres in the air. Set up and solve a system of equations to determine the quadratic relation, of the form h(t) at bt c, that models this situation. Use this relation to determine the height, h(t), of the arrow at t0 seconds.

4 Topic 4: Graphing transformations of y cos and y sin. Graph the following relation: 009 Find the mapping rule: (,y) ( + 90, y ) Make two tables. y sin y y Graph the following relation: ( y ) sin 90 Find the mapping rule: : (,y) ( 90, y ) Make two tables. y sin y y

5 Try: 4 y. Graph the following relation: cos 90 Topic 5: Finding equations of cos or sin. 009 A mother puts her child on a Merry-Go-Round. To watch her child, she stands at a point that is initially metres away from the child, which is the closest distance between the mother and child. At 6 seconds, the child is 5 metres from his mother, which is the farthest distance between the mother and child. Assuming the distance between the mother and child varies sinusoidally with time, determine the relation that models this situation. Find the following information: ma min amplitude 5 6 distance (m) sinusoidal ais: y y 9 ma +min Horizontal stretch: period Horizontal Translation: For cosine it is 6 or 8 right. For sine it is or 5 or 7 right. Mapping rule: : (, y) (a ± b, cy ± d) Horizontal stretch a Horizontal Translation b Vertical Stretch (amplitude) c Vertical Translation (sinusoidal ais): y d Reflection if c<0 time (seconds) Cosine: OR Sine:, y + 6, 6y + 9 0, y ( +,6y + 9) 0 Equation: 6 y 9 cos0( 6) Equation: 6 y 9 sin0( )

6 Try: Jack is riding a Ferris wheel. The graph below shows his height above the ground with relation to time. Determine the equation of the function in terms of sine or cosine that describes Jack s height above the ground in relation to time. height(m) time(s) Topic 6: Simplifying rational epressions 009 Simplify the following epression: Factor where possible Find the common denominator (+) + (+) + (+) the (+)s cancel out

7 008 Simplify the epression : Factor where possible + ( ) (+5) + (+) +5 (+) Change to multiplication and write the reciprocal + ( ) (+5) +5 (+) + (+) Cancel out like terms ( ) Try:. Simplify the following epression.. Simplify the following epression Topic 7: Trigonometric Proofs 009 sec sin cos Prove: Left Side Right Side sec θ( sin θ) cos θ cos θ (cos θ) cos θ cos θ cos θ cos θ cos θ cos θ 008 Prove : cos sin sin csc Left Side + cos θ Right Side csc θ sin θ + cos θ sin θ + cos θ

8 Try:. Prove: sec sec sin. Prove that cot sec sin cos sec Topic 8: Simplifying trigonometric epressions using eact values. 009 Simplify the following epression, epressing your answer in EXACT simplest form: sin0 cos45 cos40 sin Find the EXACT value for the following epression : 008 tan(60 ) cot(60 ) Try:. Simplify the following epression, epressing. Simplify, epressing your answer in eact your answer in EXACT simplest form: radical : sin 5cos 5 cos 0 cos 45 cos 0 + sin 45 sin 0

9 Topic 9: Solving trigonometric equations. 009 sin sin 0 where Solve Set the two factors equal to 0 and solve for θ. 0 0 You can use your calculator to help here but you should know when and. Using a calculator θ sin and θ sin θ 0 θ 90 0 θ < 60 means you must find all values of θ between 0 and 60 that make and. There is one more answer for. Sin is also equal to in the nd quadrant at θ 50. So there are three answers: θ 0, 50, and 90 You can use your calculator to check these: sin 0, sin 50, sin 90. Solve cos 0 where Solve for θ. cos θ + 0 cos θ cos θ You can use a calculator to find one answer : θ cos θ 50 Cos is also equal to in the rd quadrant at θ 0. You can use your calculator to check theses: cos and cos Try:. Solve sin 0 where Solve the following equation for : 4cos

10 Topic 0: Confidence Intervals 009 Good Day Tires claims that their new line of tires will last for km. A consumer research group decides to test this claim. The group randomly selects 400 tires and tests each tire. The data from this sample shows that the mean life of a tire is km, with a standard deviation of 500 km. a) Algebraically determine the 95% confidence interval for the mean life of a tire. Formula for 95% confidence interval is n where z.96 sample mean z S S n sample standard deviation sample size The 95% confidence interval is: 7785 to 7847 b) Does the confidence interval support the claim of Good Day Tires? Eplain your answer. The confidence interval does not support the claim of Good Day Tires because the confidence interval does not capture the claim of km. 008 A factory has 00 workers with a mean weekly salary of $650. A simple random sample of 50 of these workers is selected. The sample has a mean weekly salary of $60 with a standard deviation $40. Algebraically calculate the 95% confidence interval. Formula for 95% confidence interval is where z.96 n sample mean 60 z S S n sample standard deviation sample size The 95% confidence interval is: to Try: A soft drink manufacturer advertises that a certain size of its diet cola has a mean weight of 59 grams. A random sample of 50 bottles is selected. The sample mean is 589. grams and the sample standard deviation is.7 grams. Algebraically determine the 95% confidence interval. Does the confidence interval support the company s claim?

11 Topic : The Normal Distribution (Bell Curve) 009 The number of chocolate chips in a certain brand of chocolate chip cookies is known to be normally distributed with a mean of 5 chips per cookie and a standard deviation of.5. Draw and label a normal distribution curve for this situation, and determine the percentage of cookies has between and 6.5 chocolate chips. Sketch and label the normal. curve. Between and 6.5:.5% + 4% + 4% 8.5% 008 The speed limit for vehicles crossing the Pinware River bridge is 60 km/h. The speed of vehicles crossing the bridge is normally distributed with an average speed of 68 km/h and a standard deviation of 8 km/h. Draw and label a normal distribution curve and determine what percent of the vehicles are travelling over the speed limit of 60 km/h. Sketch and label the normal. Over 60 km/h: 4% + 4% +.5% +.5% 84% Try: The life epectancy of a certain brand of hair dryer is known to be normally distributed with a mean of years and a standard deviation of 0.75 years. A company produces 000 of these hair dryers. Draw and label a normal distribution curve and determine how many hair dryers are epected to last more than.5 years.

12 Topic : Law of Sines, Law of Cosines, Area of a Triangle, and Right Triangle Trigonometry 009 Algebraically determine the measure of given that is obtuse. 57 Use right triangle trigonometry to find. (Law of sines could also be used.) sin 57..cm. sin Use Law of Sines to find θ sin sin sin.4 θ sin. 70 sin X y sin Y z or sin X sin Z sin Y y sin Z z θ 45 However, θ is obtuse so θ A triangular sign needs to be painted. The sides measure 60cm, 70cm, and 00cm. a) Determine the value of and the area of the triangle. Use Law of Cosines to find θ. a b + c bc cos θ cos θ cos θ cos θ cos θ cosθ 500 cos θ cos θ θ cos.786 θ 00 70cm 00cm 60cm b) If a can of paint costs $7.50 and a can of paint cover 50cm², determine the cost to paint the sign assuming you must purchase entire cans (no partial cans). Use the are formula: Area ab sin C Area sin cm Cans of paint Cost $

13 Try:. From two drilling platforms A and B, 50 km apart on the Grand Banks, a fishing vessel, S, is sighted. If SAB 5 and SBA 5, find the distance from the vessel to the nearest platform. B 5 0 k m 5 A 5 S. A soccer net is 8 m wide. If a player is 5 m from the closer post and m from the farthest post, what is the angle,, in which he must kick the ball in order to score? 8 m m 5 m