Answers and Explanations 283
Answers
401–500
Finally, substitute in the value of x = 46
180
π radians (that is, 46°) to find the estimate of
tan 46°:
L 46
180
1 2 46
180 4
1 2 46
180
45
180
1 2 180
1 9
π
π π
π
π
π
π
( ) (
)
(
)
( )
= +
−
= +
−
= +
= + 0 0
1 035
≈ .
432.
4
2
π r dr
dt
The volume of a sphere is V
r
= 4
3
3
π . To find the rate of expansion, take the derivative
of each side with respect to time:
V
r
dV
dt
r dr
dt
r dr
dt
=
=
=
4
3
4
3
3
4
3
2
2
π
π
π
( )
433.
8π m
2
/s
The area of a circle is A = πr
2
. To find the rate of increase, take the derivative of each
side with respect to time:
A
r
dA
dt
r dr
dt
r dr
dt
=
=
=
π
π
π
2
2
2
( )
The circle increases at a rate of dr
dt
= 1 meter per second, so when radius r = 4 meters,
the area increases at the following rate:
dA
dt
=
=
2 4 1 8
π
π
( )( )
m /s
2
434.
508
Take the derivative of both sides with respect to t:
y x
x
x
dy
dt
x dx
dt
x dx
dt
dx
dt
=
+
+
=
+
+
4
2
3
3
4
6
1
Then substitute in the given values, x = 3 and dx
dt
= 4:
dy
dt
=
( ) +
+
=
4 3 4 6 3 4 1 4
508
3
( )
( )( ) ( )