Applied Mathematics 10 Extra Practice ExercisesÑ · PDF fileEXTRA PRACTICE EXERCISES...

EXTRA PRACTICE EXERCISES 117

A D D I S O N - W E S L E Y

Applied Mathematics 10 Extra Practice Exercises— Chapter 7

TABLE OF CONTENTS ??HELP


Applied Mathematics 10 Extra Practice Exercises— Chapter 7


TUTO R IAL 7 .1: Indirect Measurement

1. With respect to 4TOM, identify each of these parts.

a) the side opposite ∠M b) the side adjacent to ∠Mc) the side adjacent to ∠O d) the hypotenusee) cos ∠M f) tan ∠Mg) sin ∠O h) cos∠Oi) Pythagorean theorem

2 . Determine the missing measures in each right triangle. Round sidemeasures to one decimal place and angle measures to the nearest degree.a) b)

c) d)

3 . The vertical distance between floors at a department store is 10 m.An escalator that has an angle of inclination of 26˚ connects twofloors. How long is the escalator?

4 . On a downhill portion of a railway track that is 100 m long, thealtitude of the train decreases by 5 m. Determine x, the angle ofdepression of the track.

5 m100 mx˚

10 m

26˚

C O

W

w

1528

I

G24.4

15.1

P p˚

A

c

T

10

C43˚

O D

16d

G

56˚

O

MT

For exercises 5 to 8, draw a diagram to represent the situation. Round allanswers to two decimal places.

5. A guy wire is attached to a telephone pole and anchored to theground at a point 4 m from the base of the pole. The angle ofinclination of the wire is 55˚.a) How far above the ground on the telephone pole is the

wire attached?b) What is the length of the guy wire?

6 . A wheelchair ramp is 8.2 m long. It rises 0.94 m. Determine theangle of inclination of the ramp.

7 . A television tower is 30 m high. a) How long is the shadow when the sun is at an angle of elevation

of 60˚?b) How long is the shadow when the sun is at an angle of elevation

of 45˚?

8 . On a straight, 120 m section of a ski slope, the angle of depression is 15˚.a) What horizontal distance does the ski run cover?b) What vertical distance does the ski run cover?c) What is the slope of the ski run?

9 . At a point 28 m from a building, the angle of elevation to the top ofthe building is 65˚. The observer’s eyes are 1.5 m above the ground.How tall is the building?

10 . Grain is stored in a cone-shaped pile. The vertex angle of the conehas an angle measure of 110˚ and the base of the pile has a radius of15 m. Determine the height of the grain pile.

15 m

110˚

28 m1.5 m

65˚

118 EXTRA PRACTICE EXERCISES


Applied Mathematics 10 Extra Practice Exercises—Chapter 7


11. A pilot is flying from Hudson Bay to La Ronge. The towns are 489 km apart. On a map, La Ronge is 314 km west and 375 km north of Hudson Bay. How many degrees west of north must the pilot direct his plane to reach La Ronge? Give your answer to thenearest degree.

Answers: 1. a) TO b) MT c) TO d) OM e) MTOM f) TO

MT g) MTOM h) TO

OMi) OM2 = TO2 + MT2 2. a) 13.3 b) 10.7 c) 52˚ d) 23.6 3. 23 m 4. 2.87˚ 5. a) 5.71 m b) 6.97 m 6. 6.58˚ 7. a) 17.32 m b) 30 m 8. a) 115.91 m b) 31.06 mc) –0.27 9. 62 m 10. 11 m 11. 40˚

TUTORIAL 7.2: Solving Problems Using More ThanOne Right Triangle

For this exercise set, express all lengths to two decimal places and allangles to the nearest tenth. Diagrams are not drawn to scale.

1. Calculate the length of BC.

2. Calculate the measure of ∠ABC.

3. Two office towers are 65 m apart. The angle of depression from thetop of the shorter tower to the base of the taller tower is 68˚. Theangle of elevation from the same point to the top of the taller tower is 22˚.

3

C

2

A

B 5

50˚60˚

CB

A

8 cm

W

N

E

S

314 km

375 km

489 km

La Ronge

Hudson Bay





a) Calculate x and y.b) What is the height of the taller tower?

4. Two office towers are 30 m apart. From the top of the shorter tower,the angle of elevation to the top of the other tower, which is 250 mhigh, is 70˚.

a) How much taller is the taller tower?b) Determine the height of the shorter tower.c) Determine the angle of depression to the base of the taller tower

from the top of the shorter tower.

5. Two cabins, A and C are located a distance apart on the bank of ariver. On the other side of the river from the two cabins is aboathouse, B. It is 420 m from cabin C to the boathouse and theangle at cabin C between the boathouse and cabin A is 15˚. Fromcabin A, the angle between cabin C and the boathouse is 70˚.

a) Determine x.b) How far is cabin A from the boathouse?c) How far apart are the cabins?

15˚ 70˚

B

A

420 m

C

x

250 m

30 m

70˚

65 m

68˚

22˚ x

y





6. An ocean freighter sends a distress signal. A coast guard cutter that is50 nautical miles west and 15 nautical miles south of the freighterhears the distress call. A second freighter that is 20 nautical mileseast and 45 nautical miles south of the freighter in distress also hearsthe call.

a) Which vessel is closer to the ship in distress and by how much?b) At what angle from north would this vessel have to head to reach

the ocean freighter?

7. A non-standard roof on a house has one side 18 m long and the otherside is 23 m long. The peak is 14 m high.

a) Determine the measure of the angle formed at the peak.b) What is the span of the roof?

8. From a point on the west bank of a river 2 km wide, two speedboatsleave for their respective cabins on the east side of the river. Thedistance to the closer cabin is 3.5 km and the distance to the furthercabin is 4.2 km. What is the measure of the angle between the boats’ paths?

2 km

3.5

km

4.2 km

18 m 23 m14 m

N

E

S

2050

15

45Coast Guard Cutter

Freighter

Freighterin DistressW





9. Olivia looks out the window of her apartment building and sees aCorvette down the street at an angle of depression of 18˚. A littlefarther down the street, she sees a police car at an angle of depressionof 15˚. Her apartment window is 35 m above street level. How farapart are the Corvette and the police car?

10. For the gorge in the diagram below, calculate:a) the width (w)b) the depth (d)

11. From the top of a 100 m tower, a fire ranger spots two fires. One fireis due east of the tower at an angle of depression of 16˚. The other isdue west and has an angle of depression of 23˚. Calculate thedistance between the fires.

Answers: 1) 23 cm 2. 53˚ 3. a) x = 26.26 m; y = 160.88 m b) 187.14 m 4. a) 82.42 m b) 167.58 m c) 79.9˚ 5. a) 108.70 m b) 115.68 m c) 445.25 m 6. a) The freighter is closer by 2.96 nautical miles. b) 24.0˚ 7. a) 91.4˚ b) 29.56 m8. 116.8˚ 9. 23 m 10. a) 32 m b) 24 m 11. 584 m

100 m

23˚ 16˚

63 mw

d

27˚

37˚

35 m

18˚

15˚





TUTORIAL 7.3: Extending the Concepts of Sine and Cosine for Angles from 0˚ to 18 0˚

1. Without using a calculator, state the angles that are supplementary to:a) 55˚ b) 110˚ c) 137˚ d) 68˚

2. Without using a calculator, state the angles that are complementary to:a) 55˚ b) 26˚ c) 45˚ d) 100˚

3. Without using a calculator, find the angle supplementary to the onegiven and state its sine or cosine.a) sin 75˚ = 0.9659 b) sin 113˚ = 0.9205c) sin 179˚ = 0.0175 d) sin 44˚ = 0.6947e) cos 23˚ = 0.9205 f) cos 174˚ = −0.9945g) cos 45˚ = 0.7071 h) cos 93˚ = −0.0523

4. Without using a calculator, find the angle complementary to the onegiven and state its cosine.a) sin 15˚ = 0.2588 b) sin 73˚ = 0.9563c) sin 1˚ = 0.0175 d) sin 68˚ = 0.9272

5. Without using a calculator, find the angle complementary to the onegiven and state its sine.a) cos 60˚ = 0.5 b) cos 35˚ = 0.8192c) cos 71˚ = 0.3256 d) cos 24˚ = 0.9135

6. Determine the sine and cosine of each of the following angles. Writeyour answer correct to four decimal places.a) 40˚ b) 140˚ c) 111˚ d) 97˚e) 69˚ f) 56˚ g) 158˚ h) 22˚i) 24˚ j) 83˚

7. Why do 158˚ and 22˚ have the same sine ratio and opposite cosine ratios?

8. Without using a calculator, what do you know about ∠D if the cosineof ∠D is positive and the measure of ∠D is between 0˚ and 180˚?

9. Without using a calculator, what do you know about the measure of∠A if sin A = cos A and ∠A is between 0˚ and 90˚?

10. Calculate the value(s) for ∠A that satisfy each of the equations listedif the measure of ∠A is between 0˚ and 180˚. Give your answer tothe nearest degree.a) sin A = sin 150˚ b) sin A = sin 67˚c) cos A = − cos 100˚ d) cos 35˚ = − cos Ae) cos A = sin 40˚ f) sin A = cos 12˚g) sin A = 0.8361 h) sin A = 0.4111

i) sin A = 14 j) cos A = 0.5719

k) cos A = −0.7146 l) cos A = 14





Answers: 1. a) 125˚ b) 70˚ c) 43˚ d) 112˚ 2. a) 35˚ b) 64˚ c) 45˚ d) There is noangle. 3. a) 105˚; sin 105˚ = 0.9659 b) 67˚; sin 67˚ = 0.9205 c) 1˚; sin 1˚ = 0.0175d) 136˚; sin 136˚ = 0.6947 e) 157˚; cos 157˚ = −0.9205 f) 6˚; cos 6˚ = 0.9945g) 135˚; cos 135˚ = −0.7071 h) 87˚; cos 87˚ = 0.0523 4. a) 75˚; cos 75˚ = 0.2588b) 17˚; cos 17˚ = 0.9563 c) 89˚; cos 89˚ = 0.0175 d) 22˚; cos 22˚ = 0.92725. a) 30˚; sin 30˚ = 0.5 b) 55˚; sin 55˚ = 0.8192 c) 19˚; sin 19˚ = 0.3256 d) 66˚;sin 66˚ = 0.9135 6. a) sin 40˚ = 0.6428; cos 40˚ = 0.7660 b) sin 140˚ = 0.6428;cos 140˚ = −0.7660 c) sin 111˚ = 0.9336; cos 111˚ = −0.3584 d) sin 97˚ = 0.9925;sin 97˚ = −0.1219 e) sin 69˚ = 0.9336; cos 69˚ = 0.3584 f) sin 56˚ = 0.8290;cos 56˚ = 0.5592 g) sin 158˚ = 0.3746; cos 158˚ = −0.9272 h) sin 22˚ = 0.3746;cos 22˚ = 0.9272 i) sin 124˚ = 0.8290; cos 124˚ = −0.5592 j) sin 83˚ = 0.9925;cos 83˚ = 0.1219 7. They are supplementary angles. 8. ∠D is an acute angle(between 0˚ and 90˚). 9. ∠A = 45˚ 10. a) 30˚ b) 113˚ c) 80˚ d) 145˚ e) 50˚f) 78˚ g) 57˚ or 123˚ h) 24˚ or 156˚ i) 14˚ or 166˚ j) 55˚ k) 136˚ l) 76˚

TUTORIAL 7.5: Solving Problems Using the Sine Law

Diagrams are not drawn to scale.

1. Calculate the length of the indicated side in each triangle to thenearest tenth of a unit.a) b)

c)

2. Calculate the measure of the indicated angle in each triangle to thenearest degree.a) b)

c)

PN30

21

M

40˚

x

KJ

58.141.9

L

74.1˚ xBA

95

C

45˚ x

Q

P

28r

R

106˚

47˚

ZX

33.12

Y

93.4˚67.2˚

B

c

C 4

A

65˚

35˚





3. A roof has a span of 12 m. The angles of the roof are 35˚ and 20˚with the horizontal. Determine the length of both slants of the roof tothe nearest tenth of a metre.

4. A radar station is tracking a cruise ship and a freighter. The anglebetween the lines of sight to the two vessels from the radar station is100˚. The cruise ship is 4.5 km from the radar station and 6.3 kmfrom the freighter. What is the angle, to the nearest degree, between thelines of sight from the freighter to the cruise ship and the radar station?

5. Hole #6 at the Kenosee Lake Golf Course is straight from tee box togreen. Anthony hit his tee shot 265 yards but 15˚ off the path to thegreen. If he hits his next shot so that it approaches the hole at anangle of 23˚ with the straight-line path, he will be on target to sinkthe shot. How far must he hit the ball to make the shot? Answer tothe nearest yard.

6. A yacht race follows a triangular course. The yachts cover 5.4 km inthe first leg before turning at an angle of 75˚ for the second leg. Afterthe second leg, they make an 85˚ turn and head back to the start lineto finish the race. What is the total distance of the race to the nearesttenth of a kilometre?

7. Joe and Bob are standing, facing each other, 630 m apart when theysee a plane flying overhead. Using clinometers, they estimate theangles of elevation of the plane to be 52˚ and 68˚ respectively. Howfar is the plane from each person to the nearest tenth of a metre?

5.4 km75˚

85˚

Start

265 yards

Green

Tee Shot

Tee Box15˚ 23˚

6.3 km

4.5 km

RadarStation

Freighter CruiseShip

100˚

12 m20˚ 35˚





8. A tower is supported by two guy wires that are attached on the sameside of the tower and in a line with the base of the tower. The wiresare 810 m and 650 m long. The shorter wire makes an angle of 65˚with the ground. At what angle does the longer wire meet theground? Answer to the nearest degree.

9. To calculate the height of the Manitoba Legislative Buildingincluding the Golden Boy statue on top, Martin stood a distance awayand measured the angle of elevation to the top of the statue to be 37˚.He then moved 30 m farther away and found the measure of thisangle of elevation to be 30˚.

a) What is the distance from Martin to the top of the statue duringhis first measurement? Answer to 1 decimal place.

b) How far was Martin standing from the base of the building duringhis second measurement? Answer to 1 decimal place.

Answers: 1. a) 6.3 b) 30.6 c) 17.4 2. a) 23˚ b) 44˚ c) 113˚ 3. 5.0 m and 8.4 m 4. 45˚ 5. 176 yards 6. 12.5 km 7. 674.5 m from Joe, 573.2 m from Bob 8. 47˚9. a) 123.1 m b) 128.3 m

30 m37˚ 30˚

810 mTower

650 m

65˚

630 m

Plane

JoeBob 52˚ 68˚





TUTORIAL 7.6: The Cosine Law

Diagrams are not drawn to scale.

1. Calculate the length of the indicated side in each triangle to onedecimal place.a) b)

c)

2. Calculate the measure of the indicated angle in each triangle to thenearest degree.a) b)

c)

For exercises 3 to 11, round side measures to the nearest tenth and anglemeasures to the nearest degree.

3. A boat is towing two waterskiiers. One of the skiers has a towropethat is 50 feet long, and the other has a rope that is 60 feet long. Atone point in time, the angle formed by the towropes is 35˚. How farapart are the skiers at that point?

60 ft

50 ft

Boat

Skier 1

Skier 2

35˚

62.3

37.1

65.5

J M

I

x

17

15

23

N

DE x

7

3 9

U

SE

x˚

36.2

19.4

T

M

t

I70.4˚

11

7R Y

y

O

112˚

15

12N A

n

J

33˚





4. A tent is in the shape of an isosceles triangle. Each side is 1.8 m longand the angle formed between the sides is 74˚.

a) How wide is the floor of the tent?b) Each person who sleeps in the tent requires floor space

approximately 0.5 m wide. Using this information, how manypeople can sleep in the tent?

5. A triangular yard is bounded by a brick wall, a road, and a row oftrees. The wall is 110 feet long, the road is 125 feet long, and the rowof trees is 86 feet long.

a) Which angle is the smallest in the yard?b) Calculate the measure of the smallest angle.

6. Two vehicles leave Yorkton at the same time, traveling in directionsthat are 140˚ apart. One vehicle averages 105 km/h while the othertravels at an average speed of 95 km/h. How far apart are the vehiclesafter three hours? Draw a diagram.

7. Tuft’s Point is 3.1 km from Yarell’s Bay on a bearing of 102˚. A sailboat leaves Yarell’s Bay on a bearing of 138˚. After sailing for 2.5 km, the boat turns toward Tuft’s Point. How far is the sailboatfrom Tuft’s Point?

3.1 km

2.5 km

N

138˚102˚Yarell’ s

Bay

Tuft’ sPoint

Road125 ft

Trees86 ft

Brick Wall 110 ft

1.8 m1.8 m74˚





8. A square floor tile must be cut into a triangular shape to complete afloor pattern. The sides of the triangle required are 15.2 cm, 12.5 cmand 9.3 cm. At what angles must the tile be cut to accomplish this?

9. A hockey net is 2 m wide. A player receives a pass when he is 5 mfrom one goal post and 6.5 m from the other goal post.

a) Within what angle must the player shoot in order to score?b) Where is the player located for the shot: between the goal posts in

front of the net or outside of the goal posts to the side of the net?Justify your answer.

Answers: 1. a) 8.2 b) 15.1 c) 34.9 2. a) 123˚ b) 48˚ c) 68˚ 3. 34.4 feet 4.a ) 2.1 m b) 4 people 5. a) The angle between the brick wall and the road b) 42˚ 6. 563.9 km 7. 1.8 km 8. 38˚, 55˚, 87˚ 9. a) 13˚ b) One of the angles from the netto the player is 133˚ and therefore, the player is to the side of the net.

TUTORIAL 7.7: Selecting a Strategy

Diagrams are not drawn to scale. If no diagram is given, draw one torepresent the situation before completing the exercise. Express all lengthsto the nearest tenth and all angles to the nearest degree.

1. For each triangle, determine the indicated measures.a) b)

A

h

B3

C

6

34˚

x

4

A

a

B

C65˚

75˚

6.5 m

2 m

5 m

15.2 cm

12.5 cm

9.3 cm





c) d)

2. From a certain point, the angle of elevation to the top of a churchsteeple is 9˚. At a point 100 m closer to the steeple, the angle ofelevation is 15˚. Calculate the height of the steeple.

3. A tower is supported by two guy wires attached to the top of thetower and fixed to the ground on opposite sides of the tower 27 mapart. One wire is 19.3 m long and meets the ground at an angle of 53˚.a) Determine the length of the second wire.b) What angle does the second wire make with the ground?c) What is the height of the tower?

4. A triangular park has sides of length 200 m, 155 m and 172 m.

a) Determine ∠A.b) Determine h.c) Calculate the area of the park.

5. To determine the height of a cliff, a surveyor measured the angle ofelevation of the top of the cliff from a point away from the base to be45˚. He then moved 20 m further away from the base of the cliff andfound the angle of elevation to the top to be 37˚. Determine theheight of the cliff.

6. The end of a lean-to for cattle is in the shape of an obtuse triangle asshown below.

8 feet

4 feet

roof

110˚

A

B

172 m155 m

200 mC

h

15˚9˚100 m

A

B

4.1 3.1

4.5Cx

A

B

3

4 C

6

120˚





a) Determine the length of the roof.b) Determine the angle that the roof of the shed makes with

the ground.

7. A pilot is headed to another town from his home airport. He travels120 km due east; he turns at a angle of 135˚; he continues northeastfor 100km until he reaches the town.

a) What is the straight-line distance from his home airport to the town?

b) On what bearing must the pilot fly to reach his destination in theshortest possible distance?

8. In the design of a ski chalet, the slant of the roof must be steepenough for the snow to slide off. An architect originally designed theroof to span 45 feet with slanted sides of 36 ft and 30 ft.

He decided it would be better to modify the roof by increasing themeasure of the smaller angle by 10˚ thus increasing the length of theside opposite that angle.a) What is the new angle measure?b) What is the new length of this side?

Answers: 1. a) a = 5.6 b) x = 16˚ , h = 4.6 feet c) b = 6.1 d) x = 62˚ 2. 38.7 m 3. a) 21.8 m b) 45˚ c) 15.4 m 4. a) 56˚ b) 128.9 m c) 12 889 m2 5. 61.2 m 6. a) 10.1 feet b) 48˚ 7. a) 203.4 km b) 070˚ 8. a) 52˚ b) 36.4 feet

A

B

30 feet 36 feet

45 feetC

100 km

Town

Home120 km

N

135˚





Applied Mathematics 10 Extra Practice ExercisesÑ · PDF fileEXTRA PRACTICE EXERCISES...

Documents

Transcript of Applied Mathematics 10 Extra Practice ExercisesÑ · PDF fileEXTRA PRACTICE EXERCISES...