Calculus by Briggs and Cochran
Section 3.8- Related Rates
9.
A spherical balloon is inflated and its volume increases at a rate
of 15 in$ /min. What is the rate of change of its radius when the
radius is 10 in?
%
Z asphereb œ 1<$ o differentiate both sides with respect to >
$
.Z
.Z .<
.<
.<
.Z
œ
†
œ %1<#
o solve for
and replace
with 15
.>
.< .>
.>
.>
.>
.<
"&
œ
o replace < with 10
.>
%1<#
.<
"&
$
$
œ
œ
œ
.
.>
%1a"!!b
%1a#!b
)!1
15.
Ans:
$
in/min
)!1
A surface ship is moving (horizontally) in a straight line at
10 km/hr. At the same time an enemy maintains a position
directly below ship while diving at an angle that is #!° below the
horizontal. How fast is the submarine's altitude decreasing?
10 km/hr
Ship
Sub
x
20
y
Consider the right triangle with the top side of length x and
.B
a height of y. Note that
œ "! km/hr and unknown in the
.>
problem is
.C
Þ
.>
Find an equation that relates B and C that involves the #!° angle.
C
The simplest equation is >+8 #!° œ or C œ B >+8 #!°Þ
B
Now differentiate both sides of the equation with respect to >Þ
.C
. a>+8 #!° † B b
. a>+8 #!° † Bb .B
.B
œ
œ
†
œ >+8 #!° †
œ
.>
.>
.B
.>
.>
(remember .BÎ.> œ "!Ñ
œ "! >+8 #!° œ $Þ'% Þ Ans: 3.64 km/hr
21.
Sand falls from an overhead bin and accumulates in a conical
pile with a radius that is always three times its height. Suppose
the height of the pile increases at a rate of 2 cm/s when the pile
is 12 cm high. At what rate is the sand leaving the bin at that
instant?
Side View
h
r
Z aconeb œ
" #
1< 2 o we can eliminate one of the 2 variables
$
since < œ $2. Therefore
"
Z œ 1a$2b# 2 œ $12$
$
Now differentiate both sides with respect to >Þ
.Z
.
œ ˆ$12$ ‰
.>
.>
.Z
.
.2
.2
ˆ$12$ ‰ †
œ
œ *12# †
œ *1a"#b# a#b œ #&*#1Þ
.>
.2
.>
.>
Ans: #&*#1 cm$ /s
31.
A rope passing through a capstan on a dock is attached to a
boat offshore. The rope is pulled in at a constant rate of
3 ft/s and the capstan is 5 ft vertically above the water. How
fast is the boat traveling when it is 10 ft from the dock?
s
dock
boat
x
5 ft
Find an equation relating &, Bß and =Þ
The equation is B# € #& œ =# Þ Now differentiate both sides
with respect to >Þ
. #
.
ˆB € #&‰ œ ˆ=# ‰
.>
.>
Note that
Similarly
Therefore
. #
. # .B
.B
ˆB ‰ œ
ˆB ‰ †
œ #B Bw where Bw œ
Þ
.>
.B
.>
.>
. #
.=
ˆ= ‰ œ #==w where =w œ
œ • $ ft/s.
.>
.>
. #
.
ˆB € #&‰ œ ˆ=# ‰
.>
.>
w
w
#BB € ! œ #==
Ê
==w
Ê B œ
Þ
B
w
Bw is the speed of the boat. We want Bw when B œ "!Þ
We also need the value of =Þ We get = from the triangle
5
s
10
Therefore =# œ #& € "!! Ê = œ È"#& and
ˆÈ"#&‰a • $b
==w
&È& a • $b
$È&
w
B œ
œ
œ
œ •
Þ
B
"!
"!
#
Ans: The speed of the boat is
$È&
ft/s.
#