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 = 0 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 = 0 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 analyzes the relationship between the graph of the function, d(t)=2.65mathrmsin
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
=
0
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'.
Can you answer this question and give step by step and why and how to get it. Can you write it (numerical method)
Can you answer this question and give step by step and why and how to get it. Can you write it (numerical method)
There are three options for investing $1150. The first earns 10% compounded annually, the second earns 10% compounded quarterly, and the third earns 10% compounded continuously. Find equations that model each investment growth and
use a graphing utility to graph each model in the same viewing window over a 20-year period. Use the graph to determine which investment yields the highest return after 20 years. What are the differences in earnings among the three
investment?
STEP 1: The formula for compound interest is
A =
nt
= P(1 + − − ) n²,
where n is the number of compoundings per year, t is the number of years, r is the interest rate, P is the principal, and A is the amount (balance) after t years. For continuous compounding, the formula reduces to
A = Pert
Find r and n for each model, and use these values to write A in terms of t for each case.
Annual Model
r=0.10
A = Y(t) = 1150 (1.10)*
n = 1
Quarterly Model
r = 0.10
n = 4
A = Q(t) = 1150(1.025) 4t
Continuous Model
r=0.10
A = C(t) =…
University Calculus: Early Transcendentals (4th Edition)
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