The duration of daylight and darkness varies during the year due to the angle of the Sun in the sky. The model d ( t ) = 2.65 sin ( 0.51 t − 1.32 ) + 12 approximates the amount of daylight d ( t ) (in hours) for Sacramento, California, as a function of the time t (in months) after January 1 for a recent year; that is, t = 0 is January 1, t = 1 is February 1, and so on. The model y = n ( t ) represents the amount of darkness as a function of t . a. Describe the relationship between the graphs of the functions and the line y = 12 . b. Use the result of part (a) and a transformation of y = d ( t ) to write an equation representing n as a function of t . c. What do the points of intersection of the two graphs represent? d. What do the relative minima and relative maxima of the graphs represent? e. What does T ( t ) = d ( t ) + n ( t ) represent?
The duration of daylight and darkness varies during the year due to the angle of the Sun in the sky. The model d ( t ) = 2.65 sin ( 0.51 t − 1.32 ) + 12 approximates the amount of daylight d ( t ) (in hours) for Sacramento, California, as a function of the time t (in months) after January 1 for a recent year; that is, t = 0 is January 1, t = 1 is February 1, and so on. The model y = n ( t ) represents the amount of darkness as a function of t . a. Describe the relationship between the graphs of the functions and the line y = 12 . b. Use the result of part (a) and a transformation of y = d ( t ) to write an equation representing n as a function of t . c. What do the points of intersection of the two graphs represent? d. What do the relative minima and relative maxima of the graphs represent? e. What does T ( t ) = d ( t ) + n ( t ) represent?
Solution Summary: The author describes the relationship between the graphs of the functions and the line y=12.
The duration of daylight and darkness varies during the year due to the angle of the Sun in the sky. The model
d
(
t
)
=
2.65
sin
(
0.51
t
−
1.32
)
+
12
approximates the amount of daylight d(t) (in hours) for Sacramento, California, as a function of the time t (in months) after January 1 for a recent year; that is,
t
=
0
is January 1,
t
=
1
is February 1, and so on. The model
y
=
n
(
t
)
represents the amount of darkness as a function of t.
a. Describe the relationship between the graphs of the functions and the line
y
=
12
. b. Use the result of part (a) and a transformation of
y
=
d
(
t
)
to write an equation representing n as a function of t. c. What do the points of intersection of the two graphs represent? d. What do the relative minima and relative maxima of the graphs represent? e. What does
T
(
t
)
=
d
(
t
)
+
n
(
t
)
represent?
Definition Definition Highest point, either on the entire domain or on the given range of a function. The plural form of 'maximum' is 'maxima'.
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