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In document RIGHT TRIANGLE RATIOS (Page 38-44)

■ ■ Solving Right Triangles

EXERCISE 1.4 Figure for

1820 Horizontal line x 70.0 m Vertical cliff Figure for 12 h b a Vertical spotlight Matched Problem 4 Answers to Matched Problems Applications

A

8.0 m 61

2. Construction In Problem 1, how far is the foot of the ladder from the wall of the building?

3. Surveying When the angle of elevation of the sun is 58°, the shadow cast by the tree is 28 ft long. How tall is the tree?

4. Surveying Find the angle of elevation of the sun at the moment when a 21 m flag pole casts a 14 m shadow.

5. Flight An airplane flying at an altitude of 5,100 ft is 12,000 feet from an airport. What is the angle of eleva- tion of the plane?

6. Flight The angle of elevation of a kite at the end of a 400.0 ft string is 28.5°. How high is the kite?

7. Boat Safety Use the information in the figure to find the distance x from the boat to the base of the cliff.

8. Boat Safety In Problem 7, how far is the boat from the top of the cliff?

9. Geography on the Moon Find the depth of the moon crater in Problem 25, Exercise 1.2.

10. Geography on the Moon Find the height of the mountain on the moon in Problem 26, Exercise 1.2.

11. Flight Safety A glider is flying at an altitude of 8,240 m. The angle of depression from the glider to the control tower at an airport is 1540. What is the hori- zontal distance (in kilometers) from the glider to a point directly over the tower?

12. Flight Safety The height of a cloud or fog cover over an airport can be measured as indicated in the fig- ure. Find h in meters if b 1.00km and   23.4.

13. Space Flight The figure shows the reentry flight pattern of a space shuttle. If at the beginning of the fi- nal approach the shuttle is at an altitude of 3,300 ft and its ground distance is 8,200 ft from the beginning of the landing strip, what glide angle must be used for the shuttle to touch down at the beginning of the land- ing strip?

17. Lightning Protection A grounded lightning rod on the mast of a sailboat produces a cone of safety as indi- cated in the following figure. If the top of the rod is 67.0 ft above the water, what is the diameter of the circle of safety on the water?

Horizon

Landing strip Final approach

Figure for 15

Summer solstice sun (noon)

Winter solstice sun (noon) 19 ft 27 75 Figure for 17 90.0

B

14. Space Flight If at the beginning of the final approach the shuttle in Problem 13 is at an altitude of 3,600 ft and its ground distance is 9,300 ft from the beginning of the landing strip, what glide angle must be used for the shuttle to touch down at the beginning of the land- ing strip?

15. Architecture An architect who is designing a two- story house in a city with a 40N latitude wishes to control sun exposure on a south-facing wall. Consult- ing an architectural standards reference book, she finds that at this latitude the noon summer solstice sun has a sun angle of 75 and the noon winter solstice sun has a sun angle of 27 (see the figure).

(A) How much roof overhang should she provide so that at noon on the day of the summer solstice the shadow of the overhang will reach the bottom of the south-facing wall?

(B) How far down the wall will the shadow of the overhang reach at noon on the day of the winter solstice?

16. Architecture Repeat Problem 15 for a house located at 32N latitude, where the summer solstice sun angle is 82 and the winter solstice sun angle is 35.

Figure for 13

18. Lightning Protection In Problem 17, how high should the top of the lightning rod be above the water if the diameter of the circle on the water is to be 100 ft?

22. Orbiting Spacecraft Height A person in an orbiting spacecraft sights the horizon line on earth at an angle of depression . (Refer to the figure in Problem 21.) (A) Express cos  in terms of r and h.

(B) Solve the answer to part (A) for h in terms of r and .

(C) Find the height of the spacecraft h if the sighted an- gle of depression and the known radius of the earth, mi, is used.

23. Navigation Find the radius of the circle that passes through points P, A, and B in part (a) of the following fig- ure. [Hint: The central angle in a circle subtended by an arc is twice any inscribed angle subtended by the same arc — see figure (b).] If A and B are known objects on a maritime navigation chart, then a person on a boat at point P can locate the position of P on a circle on the chart by sighting the angle APB and completing the cal- culations as suggested. By repeating the procedure with another pair of known points, the position of the boat on the chart will be at an intersection point of the two circles.

r 3,960

  2414

19. Diagonal Parking To accommodate cars of most sizes, a parking space needs to contain an 18 ft by 8.0 ft rectan- gle as shown in the figure. If a diagonal parking space makes an angle of 72 with the horizontal, how long are the sides of the parallelogram that contain the rectangle?

h

r

 72 8.0 ft

18 ft

20. Diagonal Parking Repeat Problem 19 using 68 in- stead of 72.

21. Earth Radius A person in an orbiting spacecraft (see the figure) h mi above the earth sights the horizon on the earth at an angle of depression of . (Recall from geom- etry that a line tangent to a circle is perpendicular to the radius at the point of tangency.) We wish to find an ex- pression for the radius of the earth in terms of h and . (A) Express cos  in terms of r and h.

(B) Solve the answer to part (A) for r in terms of h and .

(C) Find the radius of the earth if and mi.

h 335

  2247

24. Navigation Repeat Problem 23 using 33 instead of 21 and 7.5 km instead of 6.0 mi.

25. Geography Assume the earth is a sphere (it is nearly so) and that the circumference of the earth at the equator is 24,900 mi. A parallel of latitude is a circle around the earth at a given latitude that is parallel to the equator

u 2u A B A B C P P 21 6.0 mi (a) (b) Figure for 23 Figure for 21 Figure for 19

(see the figure). Approximate the length of a parallel of latitude passing through San Francisco, which is at a lat- itude of 38N. See the figure, where  is the latitude, R is the radius of the earth, and r is the radius of the paral- lel of latitude. In general, show that if E is the length of the equator and L is the length of a parallel of latitude at a latitude , then cos L E .

Figure for 25 R Equator r u u

26. Geography Using the information in Problem 25 and the fact that the circumference of the earth at the equator is 40,100 km, determine the length of the Arctic Circle (6633N) in kilometers.

27. Precalculus: Lifeguard Problem A lifeguard sitting in a tower spots a distressed swimmer, as indicated in the figure. To get to the swimmer, the lifeguard must run some distance along the beach at rate p, enter the water, and swim at rate q to the distressed swimmer.

(B) Express the total time T it takes the lifeguard to reach the swimmer in terms of , d, c, p, and q. (C) Find T (in seconds to two decimal places) if

m, m, m/sec, and m/sec.

(D) The following table from a graphing calculator display shows the various times it takes to reach the swimmer for  from 55 to 85 (X represents  and Y1 represents

T ). Explain the behavior of T relative to . For what

value of  in the table is the total time T minimum?

q 1.7

p 5.1 c 76

d 380

  51,

Figure for 27 and 28

Distressed swimmer Water Beach Lifeguard tower u c d

(E) How far (to the nearest meter) should the lifeguard run along the shore before swimming to achieve the minimal total time estimated in part (D)?

28. Precalculus: Lifeguard Problem Refer to Problem 27. (A) Express the total distance D covered by the life- guard from the tower to the distressed swimmer in terms of d, c, and .

(B) Find D (to the nearest meter) for the values of d, c, and  in Problem 27C.

(C) Using the values for distances and rates in Problem 27C, what is the time (to two decimal places) it takes the lifeguard to get to the swimmer for the shortest distance from the lifeguard tower to the swimmer? Does going the shortest distance take the least time for the lifeguard to get to the swimmer? Explain. (See the graphing calculator table in Problem 27D.)

29. Precalculus: Pipeline An island is 4 mi offshore in a large bay. A water pipeline is to be run from a water tank on the shore to the island, as indicated in the figure.

(A) To minimize the total time to the swimmer, should the lifeguard enter the water directly or run some distance along the shore and then enter the water?

Explain your reasoning. Figure for 29 and 30 u Island Shore 10 mi 4 mi P

The pipeline costs $40,000 per mile in the ocean and $20,000 per mile on the land.

(A) Do you think that the total cost is independent of the angle  chosen, or does it depend on ? Explain. (B) Express the total cost C of the pipeline in terms

of .

(C) Find C for (to the nearest hundred dollars). (D) The following table from a graphing calculator dis- play shows the various costs for  from 15 to 45 (X represents  and Y1 represents C). Explain the behavior of C relative to . For what value of  in the table is the total cost C minimum? What is the minimum cost (to the nearest hundred dollars)?

  15

(B) Use the results in part (A) to find the height of the mountain in Problem 31.

33. Surveying From the sunroof of Janet’s apartment building, the angle of depression to the base of an office building is 51.4 and the angle of elevation to the top of the office building is 43.2 (see the figure). If the office building is 847 ft high, how far apart are the two build- ings and how high is the apartment building?

32. Surveying

(A) Using the figure, show that: h d

cot  cot 

(E) How many miles of pipe (to two decimal places) should be laid on land and how many miles placed in the water for the total cost to be minimum?

30. Precalculus: Pipeline Refer to Problem 29.

(A) Express the total length of the pipeline L in terms of .

(B) Find L (to two decimal places) for

(C) What is the cost (to the nearest hundred dollars) of the shortest pipeline from the island to the water tank? Is the shortest pipeline from the island to the water tank the least costly? Explain. (See the graph- ing calculator table in Problem 29D.)

31. Surveying Use the information in the figure to find the height y of the mountain.

tan 42  y x tan 25  y 1.0 x   35. Figure for 31 y x 42 25 1.0 km b a h d Figure for 33 847 ft 43.2 51.4 34. Surveying

(A) Using the figure, show that

h d cot  cot  h a b d Figure for 32

(B) Use the results in part (A) to find the distance be- tween the two buildings in Problem 33.

35. Precalculus: Physics In physics one can show that the velocity (v) of a ball rolling down an inclined plane (ne- glecting air resistance and friction) is given by

where g is the gravitational constant and t is time (see the figure).

v g(sin )t

Galileo (1564 – 1642) used this equation in the form

so he could determine g after measuring v experimen- tally. (There were no timing devices available then that were accurate enough to measure the velocity of a free-falling body. He had to use an inclined plane to slow the motion down, and then he was able to calcu- late an approximation for g.) Find g if at the end of 2.00 sec a ball is traveling at 11.1 ft/sec down a plane inclined at 10.0.

36. Precalculus: Physics In Problem 35 find g if at the end of 1.50 sec a ball is traveling at 12.4 ft/sec down a plane inclined at 15.0.

37. Geometry What is the altitude of an equilateral trian- gle with side 4.0 m? [An equilateral triangle has all sides (and all angles) equal.]

38. Geometry The altitude of an equilateral triangle is 5.0 cm. What is the length of a side?

39. Geometry Two sides of an isosceles triangle have length 12 and the included angle, opposite the base of the triangle, is 48°. Find the length of the base.

40. Geometry The perpendicular distance from the center of an eight-sided regular polygon (octagon) to each of the sides is 14 yd. Find the perimeter of the octagon.

41. Geometry Find the length of one side of a nine-sided regular polygon inscribed in a circle with radius 8.32 cm (see the figure).

g v (sin )t Figure for 35 u Galileo’s experiment 8.32 cm

42. Geometry What is the radius of a circle inscribed in the polygon in Problem 41? (The circle will be tangent to each side of the polygon, and the radius will be per- pendicular to the tangent line at the point of tangency.)

43. Geometry In part (a) of the figure, M and N are mid- points to the sides of a square. Find the exact value of sin . [Hint: The solution utilizes the Pythagorean the- orem, similar triangles, and the definition of sine. Some useful auxiliary lines are drawn in part (b) of the figure.]

Figure for 43 u M N D A B C u M F E N s s (a) (b)

44. Geometry Find r in the figure. The circle is tangent to all three sides of the isosceles triangle. (An isosceles tri- angle has two sides equal.) [Hint: The radius of a cir- cle and a tangent line are perpendicular at the point of tangency. Also, the altitude of the isosceles triangle will pass through the center of the circle and will divide the original triangle into two congruent triangles.]

r

40

1.78 cm Figure for 44

In document RIGHT TRIANGLE RATIOS (Page 38-44)

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