(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.
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
3)
S] Another Ferris wheel at a local fair has a radius of 13m, and is 1 m above the ground at its lowest
point. The Ferris wheel makes one full rotation every 45 seconds.
Determine a sine function model for the vertical position, h, in metres, of a rider on the wheel after a time, t, in
seconds.
Assume that at t =
Determine a:
Determine c:
(a)
Os, the rider is at the highest point. Show your work to determine each parameter
Determine k:
Equation: h(t) =
Determine d:
Precalculus: Mathematics for Calculus - 6th Edition
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.