A hiking trail is roughly in the shape of a circle with radius 2000 ft . Suppose that a rectangular coordinate system is set up with the origin at the center of the circle. Donna and Eddie start walking from point A 2000 , 0 at 220 ft/min . Eddie goes off-trail and walks directly toward the center of the circle and Donna walks along the trail counterclockwise. a. Find the coordinates of each person after 5 min . Round the coordinates to the nearest foot b. How far apart are Eddie and Donna after 5 min ? Round to the nearest foot. c. If Eddie can no longer walk because he sprains his ankle after foolishly leaving the trail, at what bearing would Donna walk to find him? Round to the nearest tenth of a degree.
A hiking trail is roughly in the shape of a circle with radius 2000 ft . Suppose that a rectangular coordinate system is set up with the origin at the center of the circle. Donna and Eddie start walking from point A 2000 , 0 at 220 ft/min . Eddie goes off-trail and walks directly toward the center of the circle and Donna walks along the trail counterclockwise. a. Find the coordinates of each person after 5 min . Round the coordinates to the nearest foot b. How far apart are Eddie and Donna after 5 min ? Round to the nearest foot. c. If Eddie can no longer walk because he sprains his ankle after foolishly leaving the trail, at what bearing would Donna walk to find him? Round to the nearest tenth of a degree.
Solution Summary: The author calculates the coordinates of Donna and Eddie after each person walks for 5min from point A.
A hiking trail is roughly in the shape of a circle with radius
2000
ft
. Suppose that a rectangular coordinate system is set up with the origin at the center of the circle. Donna and Eddie start walking from point
A
2000
,
0
at
220
ft/min
. Eddie goes off-trail and walks directly toward the center of the circle and Donna walks along the trail counterclockwise.
a. Find the coordinates of each person after
5
min
. Round the coordinates to the nearest foot
b. How far apart are Eddie and Donna after
5
min
? Round to the nearest foot.
c. If Eddie can no longer walk because he sprains his ankle after foolishly leaving the trail, at what bearing would Donna walk to find him? Round to the nearest tenth of a degree.
System that uses coordinates to uniquely determine the position of points. The most common coordinate system is the Cartesian system, where points are given by distance along a horizontal x-axis and vertical y-axis from the origin. A polar coordinate system locates a point by its direction relative to a reference direction and its distance from a given point. In three dimensions, it leads to cylindrical and spherical coordinates.
1. Show that the vector field
F(x, y, z)
=
(2x sin ye³)ix² cos yj + (3xe³ +5)k
satisfies the necessary conditions for a conservative vector field, and find a potential function for
F.
1. Newton's Law of Gravitation (an example of an inverse square law) states that the magnitude
of the gravitational force between two objects with masses m and M is
|F|
mMG
|r|2
where r is the distance between the objects, and G is the gravitational constant. Assume that the
object with mass M is located at the origin in R³. Then, the gravitational force field acting on
the object at the point r = (x, y, z) is given by
F(x, y, z) =
mMG
r3
r.
mMG
mMG
Show that the scalar vector field f(x, y, z) =
=
is a potential function for
r
√√x² + y² .
Fi.e. show that F = Vf.
Remark: f is the negative of the physical potential energy, because F = -V(-ƒ).
2. Suppose f(x) = 3x² - 5x. Show all your work for the problems below.
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