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51 39 m r 37 ° south of east _____________________________________________________________________________________________ 13. SSM WWW REASONING The shortest distance between the two towns is along the
line that joins them. This distance, h, is the hypotenuse of a right triangle whose other sides
are ho = 35.0 km and ha = 72.0 km, as shown in the figure below.
SOLUTION The angle θ is given by tan θ = ho / ha
so that
−1 35.0 km θ = tan = 25.9° S of W
72.0 km W θ
h h θ We can then use the Pythagorean theorem to find h. ha
S 2
h = ho + ha2 = (35.0 km) 2 + ( 72 .0 km) 2 = 80.1 km o 8 INTRODUCTION AND MATHEMATICAL CONCEPTS 14. REASONING When the monkey has climbed as far up the
pole as it can, its leash is taut, making a straight line from the
stake to the monkey, that is, L = 3.40 m long. The leash is
the hypotenuse of a right triangle, and the other sides are a
line drawn from the stake to the base of the pole
(d = 3.00 m), and a line from the base of the pole to the
monkey (height = h). Stake
L
h
d SOLUTION These three lengths are related by the Pythagorean theorem (Equation 1.7): h 2 + d 2 = L2
h = L2 − d 2 = 15. or h 2 = L2 − d 2 ( 3.40 m )2 − ( 3.00 m )2 = 1.6 m REASONING Using the Pythagorean theorem (Equation 1.7), we find that the relation
between the length D of the diagonal of the square (which is also the diameter of the circle)
and the length L of one side of the square is D = L2 + L2 = 2 L .
SOLUTION Using the above relation, we have D 0.35 m
=
= 0.25 m
2
2
______________________________________________________________________________
D = 2L or L= 16. REASONING In both parts of the drawing the line of sight, the horizontal dashed line, and
the vertical form a right triangle. The angles θa = 35.0° and θb = 38.0° at which the person’s
line of sight rises above the horizontal are known, as is the horizontal distance d = 85.0 m
from the building. The unknown vertical sides of the right triangles correspond,
respectively, to the heights Ha and Hb of the bottom and top of the antenna relative to the
person’s eyes. The antenna’s height H is the difference between Hb and Ha: H = H b − H a .
The horizontal side d of the triangle is adjacent to the angles θa and θb, while the vertical sides Ha and Hb are opposite these angles. Thus, in either triangle, the angle θ is related to h
the horizontal and vertical sides by Equation 1.3 tan θ = o :
ha H
tan θa = a
d
tan θ b = Hb
d (1) (2) Chapter 1 Problems 9 H
Hb Ha θa θb
d d (a) (b) SOLUTION Solving Equations (1) and (2) for the heights of the bottom and top of the
antenna relative to the person’s eyes, we find that
H a = d tan θa H b = d tan θ b and The height of the antenna is the difference between these two values: H = H b − H a = d tan θb − d tan θa = d ( tan θb − tan θa ) ( ) H = (85.0 m ) tan 38.0o − tan 35.0o = 6.9 m 17. REASONING AND SOLUTION Consider the following views of the cube. Bottom
View Side View Na L c
a Na a C a Na C L Na The length, L, of the diagonal of the bottom face of the cube can be found using the
Pythagorean theorem to be
2 2 2 2 2 L = a + a = 2(0.281 nm) = 0.158 nm or L = 0.397 nm 10 INTRODUCTION AND MATHEMATICAL CONCEPTS The required distance c is also found using the Pythagorean theorem.
2 2 2 2 2 2 c = L + a = (0.397 nm) + (0.281 nm) = 0.237 nm
Then,
c = 0.487 nm 18. REASONING The drawing shows the heights of the two
balloonists and the horizontal distance x between them.
Also shown in dashed lines is a right triangle, one angle of
which is 13.3°. Note that the side adjacent to the 13.3°
angle is the horizontal distance x, while the side opposite
the angle is the distance between the two heights, 61.0 m −
48.2 m. Since we know the angle and the length of one side
of the right triangle, we can use trigonometry to find the
length of the other side. 13.3° x 61.0 m 48.2 m SOLUTION The definition of the tangent function, Equation 1.3, can be used to find the
horizontal distance x, since the angle and the length of the opposite side are known:
tan13.3° = length of opposite side
length of adjacent side (= x ) Solving for x gives x= length of opposite side 61.0 m − 48.2 m
=
= 54.1 m
tan13.3°
tan13.3° ____________________________________________________________________________________________ 19. REASONING Note from the drawing that the shaded
right triangle contains the angle θ , the side opposite the
angle (length = 0.281 nm), and the side adjacent to the
angle (length = L). If the length L can be determined, we
can use trigonometry to find θ. The bottom face of the
cube is a square whose diagonal has a length L. This
length can be found from the Pythagorean theorem,
since the lengths of the two sides of the square are
known. 0.281 nm θ
0.281 nm 0.281 nm
L SOLUTION The angle can be obtained from the inverse tangent function, Equation 1.6, as
θ = tan −1 ( 0.281 nm ) / L . Since L is the length of the hypotenuse of a right triangle whose sides have lengths of 0.281 n...
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This note was uploaded on 04/30/2013 for the course PHYS 1100 and 2 taught by Professor Chastain during the Spring '13 term at LSU.
 Spring '13
 CHASTAIN
 Physics, The Lottery

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