(c) Use the tangent line approximation at
H=1
to estimate the value of
H
that will yield a spherical
segment having volume 25 cubic inches:

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1/5 1/5 1/5 1/5 1/5 5/5
In the triangle pictured, let
A, B, C
be the angles at the three vertices, and let
a,b,c
be the sides opposite
those angles.
According to the ``law of sines,'' you always have:

a
=
sin
B
sin
A
Suppose that
a
and
b
are pieces of metal which are hinged at
C
. At first the angle
A
is
π
/4 radians=45
and the angle
B
is
π
/3 radians = 60
o
. You then widen
A
to 46
o
, without changing the sides
a
and
b
goal in this problem is to use the tangent line approximation to estimate the angle
B
(a) Notice that the angle
B
is a function of the angle
A
; i.e.
B=f(A)
. Consequently, it makes sense to
calculate the implicit derivative:
o
. Our
.

(b) Calculate
dB
dA
when
A
=
π
/4 and
B
=
π
/3; leave your answer in EXACT FORM:

(c) Write the linear approximation of
f
at
A
=
π
/4:
$$√3

(
A
–
π
/4) +
$$π3
(d) Using (c), when
A
= 46
o
,
B
is approximately
$$61.732
degrees; either answer exactly or to three decimal places.

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A metal rod 1 meter in length is placed horizontally on the
x
-axis, with its ends located at the points
P
(0,0) and
Q
= (1,0). A second long metal rod is attached to the first one at the point
Q
=(1,0) and makes an
angle of 45
o
with the first. A third long metal rod is attached to the first rod at the point
P
=(0,0) and is
free to rotate about
P
. Thus, the angle
θ
made by the first and third bars is free to change. For
θ
between
0
o
and 90
o
the second and third bars cross at a point
R
=(x,y) (see figure).
(a) Find a relationship between
x, y
and
θ
; then eliminate
y
by expressing
y
in terms of
x
. This gives you a
relationship between
x
and
θ
of the form:
(Write theta to represent
θ
.)
=
.

(b) Notice that
x=0.5
meter when
θ
= 45
o
. By approximately how many radians should you increase
θ
you want the
x
coordinate of the point
R
to decrease to
x = 0.45
meters? Use the tangent line
if
approximation.20.1
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