** ■ ■ 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