Download SECTION 5-5 Solving Right Triangles*

390
5 Trigonometric Functions
42. Air Safety. If a jet airliner climbs at an angle of 15°30 with
a constant speed of 315 miles per hour, how long will it take
(to the nearest minute) to reach an altitude of 8.00 miles?
Assume there is no wind.
★ 43.
★ 44.
★ 45.
★ 46.
Resort
Island
Astronomy. Find the diameter of the moon (to the nearest
mile) if at 239,000 miles from Earth it produces an angle of
32 relative to an observer on Earth.
Astronomy. If the sun is 93,000,000 miles from Earth and
its diameter is opposite an angle of 32 relative to an observer on Earth, what is the diameter of the sun (to 2 significant digits)?
3 miles
City
(A) Referring to the figure, show that the cost in terms of is given by
Geometry. If a circle of radius 4 centimeters has a chord of
length 3 centimeters, find the central angle that is opposite
this chord (to the nearest degree).
C() 75,000 sec 45,000 tan 300,000
Geometry. Find the length of one side of a nine-sided regular polygon inscribed in a circle of radius 4.06 inches.
(B) Calculate a table of costs, each cost to the nearest dollar, for the following values of : 10°, 20°, 30°, 40°,
and 50°. (Notice how the costs vary with . In a course
in calculus, students are asked to find so that the cost
is minimized.)
47. Physics. In a course in physics it is shown that the velocity
v of a ball rolling down an inclined plane (neglecting air resistance and friction) is given by
v gt sin ★ 50.
where g is a gravitational constant (acceleration due to
gravity), t is time, and is the angle of inclination of the
plane (see the figure). Galileo (1564–1642) used this equation in the form
g
—Cost Analysis. Refer to Problem 49. SupEngineering—
pose the island is 4 miles offshore and the cost of running
the cable along the shore is $20,000 per mile and underwater, $30,000 per mile.
(A) Referring to the figure for Problem 49 with appropriate
changes, show that the cost in terms of is given by
C() 120,000 sec 80,000 tan 400,000
v
t sin to estimate g after measuring v experimentally. (At that
time, no timing devices existed to measure the velocity of a
free-falling body, so Galileo used the inclined plane to slow
the motion down.) A steel ball is rolled down a glass plane
inclined at 8.0°. Approximate g to one decimal place if at
the end of 3.0 seconds the ball has a measured velocity of
4.1 meters per second.
Shore
20 miles
(B) Calculate a table of costs, each cost to the nearest dollar, for the following values of : 10°, 20°, 30°, 40°,
and 50°.
★★ 51.
Geometry. Find r in the accompanying figure (to 2 significant digits) so that the circle is tangent to all three sides of
the isosceles triangle. [Hint: The radius of a circle is perpendicular to a tangent line at the point of tangency.]
r
30
2.0 meters
48. Physics. Refer to Problem 47. A steel ball is rolled down a
glass plane inclined at 4.0°. Approximate g to one decimal
place if at the end of 4.0 seconds the ball has a measured
velocity of 9.0 feet per second.
★ 49.
—Cost Analysis. A cable television company
Engineering—
wishes to run a cable from a city to a resort island 3 miles
offshore. The cable is to go along the shore, then underwater to the island, as indicated in the accompanying figure. The cost of running the cable along the shore is
$15,000 per mile and underwater, $25,000 per mile.
★★ 52.
Geometry. Find r in the accompanying figure (to 2 significant digits) so that the smaller circle is tangent to the larger
circle and the two sides of the angle. [See the hint in Problem 51.]
r
30
2.0 in.