Exam 2
CalculusII, MA 112
N"-"
Exam 2 (100 points)
April 21,2004
V
K91
Box-
You may use your calculator but not maple.
(15 pts) 1-. Supposethat you are on top of an 80 meter tall building and you toss a 1.2 kg ball upwards
with a speed of 40 m/sec. Assume that there is a resistanceforce whose magnitude is 0.8
times the speed of the ball. All expressionsshould be simplified and numbers should be
written to two decimal places.
(a) Start with F : rna to determine the differential equation for o. Recall that it's easierto
write ma: F : Fn + F, and then replace oby #. Write the IVP for u in the box.
l , L M = - f \i 4 ? , t ' o , & v
-M
a
o-,k,/
A'l/, t' t'
/et
T
n+*
i;L'
I
I
Put the fVP here:
4 : . - ? , t r -, b t / v , u e ) : Y o
(b) Now solve for u. Show your work.
V(ol=/o =')
A =llt
tJ- q*-,
L?tr
{o=
t - I tl' -t q , f - . L 7 v=lt t c
,;itu
= -'17/r<
A /-frY-'674
-,6?t
e
- , b- l,Y[ ? t :/ c
-' ?7 ..[ Y
-,6rt
-/y.[ ) +c
L = trv,03
- , 6 ? v = ? ' t r+ c e ' . , ,
-,c7t
+ c e
Put your answer here: u:
-.
/ /. [ 3
+ s y . 6 3g
-,67
t
(15 pts) 2. A 500 gallon tank contains a solution of water and salt with the initial amount of salt being
10 pounds. If salt water with a concentration of 1 pound of salt per gallon enters the tank
at a rate of 4 gallons per minute and the mixture leavesthe tank at the same rate, how
much salt is in the tank at time t? Sketch the tank (vdth appropriate arrows) and set up
the differential equation. DO NOT SOLVE the DE. However, what will the amount of salt
approach as time increases?
Sketch the picture (include the arrows) and determine the DE. Let r be the amount of salt
in the tank at time t.
--tn', aL
Nal =/o
- /,
*_,9a
D0(//
v7{n';'
l7^{/w',
v
#=
t,'f a
tfl
h
Put the IVP here:
{ -
,l
6@
v
fzr
4d t: tt { - /+7 5-
DO NOT SOLVEthe DE. However,determine'limr(t) :
ilXl-t
flrr , ,(:
-!
5uO- {fo e
lvr
,r(o):/o
f
00
(15 pts) 3. Let R be the region in the first quadrant bounded by g : 12, the r axis, and the \ine r : 2.
(a). Find the volume of the solid obtained by revolving .R about the r axis. Evaluate the
integral to obtain a numerical answer.
Z*'#b*
Jh7'A'x
za,
ll
try
=
=
f
=
= [ ' n x tA r
b
toil
do
(b) Find the volume of the solid obtained by revolving R about the line r : J.
answeras an integral.
ve your
-$l
D;",,n'
t
or^tfn
[rn rl +L!";
t lx
[\ng-*r
o
G-efy
F
nL
tn
If, 0
- lnnta'
n t\{
Z nft-l-'
^
I'
r
Y
.
rr
\el
= \ n (\_/
5 - Y^J1
0
= J o '(ns - e) I l _ Y I T
Y
2
t zrT
= rr1
-
YlT
(15 pts)
. Let R be the region in the first quadrant bounded by the graphs of y : 1/i and A : r.
Use the shell method to set up the integral which gives the volume of the solid obtained by
revolving R about the line A : -1. Do not evaluatethe integral (however,the integral which
you give as your answer must be ready for maple).
2 zrr,L' rbb'
u=(
\=q
#) )''7
(v+08,
J'.n
(tf,yU
',n
(r+,)(xLi) nt
,J
12'L
,$
,1,
(15 pts) 5. Set up the i{egrgl which gives the length of the ellipse with ce4ler at the origin, major axis
is of length I along the r axis, and the minor axis is of length/along the g axis. Again set
.t tL
l- - b. r-il
#cJ)
N
up the integralbut do not evaluate. -
1*-
ry +-1):tt
fr: fc'a
s_s*
l"t**
?t t lj,
/= XA&
o e &=Lf(
vIl
[/d^=J ^ s
? r r
o
l'lp
: r,o6"
n
ca
vd
: 21",/o
t/*r: u,,rL4 rq/'u 5's-l'- W
2(3f
(15 pts) 6. A 4lb force will stretch a spring tl2 ft. How much work is done in stretching the spring 2
fb from its natural length?
4. = JL
L.=7
= ' (t ll -' *? IL
k
tl) '- .\ f X k
t()
U
o
(10 pts) 7. A tank has the shape of a paraboloid of revolution obtained by revolving A : 12 (f.or
0 ( r S 2) about the y axis. If it is full of water, how much work is required to empty the
tank by pumping all the water out over the top edge? ,{ssume r is measuredin feet and the
densiryof wateris 62.t lb/ft3. St
rc ar-l*vr!r*,
;"ft(.
1.x"
(K5r
J*^t' L6*
U > Z n *. I.,'b!l'
' t/ = f', *- (bzu1
(f-lVY
.,
u
L z vn
I
\
do
"Ju -1)
l-"'t
, ., tY
JZJI'
r
T
(
t
.
r
[
'
t
"
=
: 6t.Y
n(*
?)
=C2.{n 12
= /-o//.0/
3