(a) The accompanying figure shows a sector of radius r and central angle 2 α . Assuming that the angle α is small, use the local quadratic approximation of cos α at α = 0 to show that x ≈ r α 2 / 2. (b) Assuming that the Earth is a sphere of radius 4000 mi, use the result in part(a) to approximate the maximum amount by which a 100 mi arc along the equator will diverge from its chord.
(a) The accompanying figure shows a sector of radius r and central angle 2 α . Assuming that the angle α is small, use the local quadratic approximation of cos α at α = 0 to show that x ≈ r α 2 / 2. (b) Assuming that the Earth is a sphere of radius 4000 mi, use the result in part(a) to approximate the maximum amount by which a 100 mi arc along the equator will diverge from its chord.
(a) The accompanying figure shows a sector of radius
r
and central angle
2
α
.
Assuming that the angle
α
is small, use the local quadratic approximation of
cos
α
at
α
=
0
to show that
x
≈
r
α
2
/
2.
(b) Assuming that the Earth is a sphere of radius 4000 mi, use the result in part(a) to approximate the maximum amount by which a 100 mi arc along the equator will diverge from its chord.
At a certain ocean bay, the maximum height of the water is 4 m above i
mean sea level at 8:00 a.m. The height is at a maximum again at 8:24
p.m. The lowest sea level is 4 m below mean sea level.
a) Assuming that the relationship between the height, h, in metres,
and the time, t, in hours, is sinusoidal, determine an equation to
model the height of the water with respect to time.
b) Determine the height of the water above mean sea level at 10:00
a.m.
As the wheel of radius r cm in the figure rotates, the rod of length L attached to point P
drives a piston back and forth in a straight line. Let x be the distance from the origin to
point at the end of the rod as shown.
(a) Use the Pythagorean Theorem to show that
L² = (x − r cos 0)² + ² sin² 0.
(b) Differentiate the equation in part (a) with respect to t to show that
0=2(x-r cos 0) (d+rsin 0df)+2r² sin cos de.
dt
(c) Calculate the speed of the piston when , assuming that r = 10 cm, L = 30 cm,
and the wheel rotates at 4 revolutions per minute.
L
X
=
Piston moves
back and forth
e
Precalculus: Mathematics for Calculus - 6th Edition
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