According to Kepler’s law , the planets in our solar system move in elliptical orbits around the Sun. If a planet’s closest approach to the Sun occurs at time t = 0 , then the distance r from the center of the planet to the center of the Sun at some later time t can be determined from the equation r = a 1 − e cos ϕ where a is the average distance between centers, e is a positive constant that measures the “flatnessâ€� of the elliptical orbit, and ϕ is the solution of Kepler’s equation 2 π t T = ϕ − e sin ϕ in which T is the time it takes for one complete orbit of the planet. Estimate the distance from the Earth to the Sun when t = 90 days. [First find ϕ from Kepler’s equation, and then use this value of ϕ to find the distance. Use a = 150 × 10 6 km , e = 0.0167 , and T = 365 days.]
According to Kepler’s law , the planets in our solar system move in elliptical orbits around the Sun. If a planet’s closest approach to the Sun occurs at time t = 0 , then the distance r from the center of the planet to the center of the Sun at some later time t can be determined from the equation r = a 1 − e cos ϕ where a is the average distance between centers, e is a positive constant that measures the “flatnessâ€� of the elliptical orbit, and ϕ is the solution of Kepler’s equation 2 π t T = ϕ − e sin ϕ in which T is the time it takes for one complete orbit of the planet. Estimate the distance from the Earth to the Sun when t = 90 days. [First find ϕ from Kepler’s equation, and then use this value of ϕ to find the distance. Use a = 150 × 10 6 km , e = 0.0167 , and T = 365 days.]
According to Kepler’s law, the planets in our solar system move in elliptical orbits around the Sun. If a planet’s closest approach to the Sun occurs at time
t
=
0
,
then the distance
r
from the center of the planet to the center of the Sun at some later time
t
can be determined from the equation
r
=
a
1
−
e
cos
ϕ
where
a
is the average distance between centers,
e
is a positive constant that measures the “flatness� of the elliptical orbit, and
ϕ
is the solution of Kepler’s equation
2
π
t
T
=
ϕ
−
e
sin
ϕ
in which
T
is the time it takes for one complete orbit of the planet. Estimate the distance from the Earth to the Sun when
t
=
90
days. [First find
ϕ
from Kepler’s equation, and then use this value of
ϕ
to find the distance. Use
a
=
150
×
10
6
km
,
e
=
0.0167
,
and
T
=
365
days.]
A pendulum is release data point 5 meters to the right of its position at rest. Graph a cosine function that
represents the pendulum's horizontal displacement relative to its position at rest if it completes one back-and-forth
swing every π seconds.
A swimmer on a floating air mattress will follow the rise and fall of the
waves in lake. When the person is on the top of the wave her feet are 0.5m above the
average surface height of the lake. When she is in a trough her feet dig 0.5m into the
water (-0.5m). The peaks of the waves are separated by two seconds. Create an
equation of the sinusoidal function that models this movement assuming that she
starts at the average surface height of the lake and heads upwards.
Chapter 4 Solutions
Calculus Early Transcendentals, Binder Ready Version
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