MATH 152
Exam 2
Fall 1997
Version A
Solutions
Part I is multiple choice. There is no partial credit. You may not use a calculator.
Part II is work out. Show all your work. Partial credit will be given. You may use your
calculator.
Part I: Multiple Choice (5 points each)
There is no partial credit. You may not use a calculator. You have 1 hour.
1. The natural length of a certain spring is at 2 m. A force of 6 N will stretch the spring from
2 m to 4 m. How much work will it take to stretch the spring from 4 m to 6 m?
a. 18 J correctchoice
b. 24 J
c. 36 J
d. 48 J
e. 60 J
F kŸx " x 0 6 kŸ4 " 2 k3
F 3Ÿx " 2 6
6
6
3 Ÿ4 2 " 2 2 18
(a)
W ; Fdx ; 3Ÿx " 2 dx 3 Ÿx " 2 2
4
4
2
2
4
dy
x 2 " y 2 has direction field
dx
2. The differential equation
2
y1
a.
-2
-1
0
1
x
2
1
x
2
-1
-2
2
y1
b.
-2
-1
0
-1
-2
2
y1
c.
-2
-1
0
1
x
2
1
x
2
1
x
2
correctchoice
-1
-2
2
y1
d.
-2
-1
0
-1
-2
2
y1
e.
-2
-1
0
-1
-2
The slope must be zero when x 2 " y 2 0 i.e. along the 45 ( lines y ox.
(c).
This is only satisfied by
dy
Ÿ2x 1 Ÿy 1 is
3. The differential equation
dx
a. neither separable nor linear
b. separable but not linear
c. linear but not separable
d. separable and linear correctchoice
e. none of these
The equation is already separated.
dy
Ÿ2x 1 y Ÿ2x 1 .
(d)
To see that it is linear, we write it as
dx
4. Solve the differential equation
2
dy
3 x 3 with the initial condition yŸ0 2.
dx
4 y
a. x 3/4 2
b. 3 x 4 8
c. x 4/3 2
d. 2x 4/3
x 3 16 correctchoice
Integrate: y 4 x 3 C
Separate: ; 4y 3 dy ; 3x 2 dx
Use the initial condition: 2 4 0 3 C
So: C 16
4
3
and solve: y 4 x 3 16
Substitute back: y x 16
e.
4
(e)
5. If a ˜4, 2, 1 ™ and b ˜3, 2, 3 ™ then a 2b a. ˜"2, "2, "5 ™
b.
˜10, 6, 7 ™
c.
d.
e.
˜7, 4, 4 ™
˜1, 0,
correctchoice
"2 ™
˜11, 6, 5 ™
a 2b ˜4, 2, 1 ™ ˜6, 4, 6 ™ ˜10, 6, 7 ™
(b)
6. The triangle with vertices P Ÿ4, 2, 1 , Q Ÿ3, 3, 1 and R Ÿ3, 2, 3 is
a. equilateral
b. isosceles but not right correctchoice
c. right but not isosceles
d. isosceles and right
e. none of these
PQ Q " P Ÿ"1, 1, 0 PQ PR R " P Ÿ"1, 0, 2 PR QR R " Q Ÿ0, "1, 2 QR 2
5
5
(b)
7. In 3-dimensional space the equation x 2 z 2 16 describes
a. the interior of a circle of radius 4 in the xz-plane centered at the origin.
b. the region inside a paraboloid centered along the positive y-axis.
c. the region inside a cone centered along the positive y-axis.
d. the part of the xz-plane outside of a circle of radius 4 centered at the origin.
e. the interior of a cylinder of radius 4 centered on the y-axis. correctchoice
(e)
4
2
-4
0 -2
y t 2
2
4 -2
-4
-2
-4
4
8. The arclength of the curve y e 2x between x 1 and x 2 is given by
a. ;
b. ;
c. ;
2
1
e4
e2
2
1
e4
1 e 4x dx
1 2e 2x dx
1 4e 4x dx correctchoice
d. ; 2 2= 1 e 2x dx
e
2
e. ; 2=x 1 e 4x dx
1
L ; ds ; 1 dy
dx
2
dx ;
2
1
1 Ÿ2e 2x 2 dx ;
9. The curve y sinŸx , between x 1 and x 2
1 4e 4x dx
1
(c)
= , is rotated about the x-axis. The
2
surface area of the resulting surface is given by.
=/2
a. ; 2= sinŸx 1 cos 2 Ÿx dx correctchoice
1
b. ;
c. ;
=/2
1
=/2
1
1
2=x 1 cos 2 Ÿx dx
2= cosŸx 1 sin 2 Ÿx dx
d. ; 2=y cos "2 Ÿy 1 dy
0
e. ;
SA =/2
1
;
2=x 1 sin 2 Ÿx dx
=/2
1
2=Ÿradius ds ;
=/2
1
2=Ÿy 1 dy
dx
2
dx ;
=/2
1
2= sinŸx 1 cos 2 Ÿx dx
(a)
10. Find a unit vector in the direction of the vector Ÿ3, "4, 12 a.
b.
c.
d.
e.
v 12 , " 4 , 3
13 13 13
" 12 , 4 , " 3
13 13 13
" 3 , 4 , " 12 correctchoice
13 13 13
3 , " 4 , 12
13 13 13
" 4 , 12 , 4
13 13 13
v 9 16 144 13
Ÿ3, "4, 12 §
v v
v
3 , " 4 , 12
13 13 13
(c)
Part II: Work Out (10 points each)
Show all your work. Partial credit will be given.
You may use your calculator but only after 1 hour.
11. A plate in the shape of an isosceles triangle with base of 6 m and altitude of 9 m is put
in water with the base at the surface of the water and the point straight down. Find the
total force on the plate. Note the density of water is 1000 kg / m 3 and the acceleration of
gravity is 9.8 m / s 2 .
8
The slice is at height y and has length L within the triangle.
6
4
2
0 -3
-2
-1
0
x
1
2
3
By similar triangles: Ly 6 . So L 2 y.
9
3
So the Area of the slice is dA L dy 2 y dy.
3
The depth of the slice below the surface of the water is h 9 " y. So the force is
9
9
F ; >gh dA ; 1000 • 9.8 • Ÿ9 " y 2 y dy 1000 • 9.8 • 2 ; Ÿ9y " y 2 dy
0
0
3
3
y2
y3 9
2
2
1000 • 9.8 • • 9 3 • 1 " 1 793800 N
1000 • 9.8 •
9
"
3
3
2
2
3 0
3
12. Solve the differential equation
3
dy
3x 2 y 6x 2 e x with the initial condition yŸ0 5.
dx
3
dy
" 3x 2 y 6x 2 e x
dx
3
We identify the coefficients: P "3x^2 and Q 6x 2 e x
We write the equation in standard form:
; Pdx
"; 3x 2 dx
e
e "x
The integrating factor is I e
3 dy
3
" 3x 2 e "x y 6x 2
Multiply the equation by the integrating factor: e "x
dx
3
Identify the left as a total derivative: d e "x y 6x 2
dx
3
Integrate both sides: e "x y ; 6x 2 dx 2x 3 C
Use the initial condition yŸ0 5 to get e 0 5 0 C so that C 5
3
3
Substitute back to get e "x y 2x 3 5 or y Ÿ2x 3 5 e x
3
13. A water trough has the shape of half of a cylinder on its side, as shown in the figure.
The radius is 2 m and the length is 10 m. How much work does it take to pump the
water out of the top of the trough if the trough is full of water? The density of water is
1000 kg / m 3 .
-2
x
1
-1
2
0
-0.5
-1
-1.5
-2
The slice is at depth h y below the surface of the water.
The length of the slice is L 2x 2 4 " y 2
So its volume is dV 10L dy 20 4 " y 2 dy
and the weight of the slice is dF >g dV 20>g 4 " y 2 dy So the work is
3/2
2
2
W ; h dF ; 20>gy 4 " y 2 dy "10>g ; u du "10>g 2u
0
0
3
2
"10>g 2 Ÿ4 " y 2 3/2 160 >g 160 1000 • 9.8 X 522667 J
0
3
3
3
14. Find the x-coordinate of the centroid of the area bounded by the parabola y x 2 and
the lines x 1 and y 0.
1
0.8
0.6
0.4
The area is A 0.2
0
0.2
0.4
0.6
0.8
;
1
0
x 2 dx x3
3
1
0
1
3
1
x4 1 1
0
0
4
4 0
1
My
41 3
So the x-coordinate of the centroid is x 4
A
3
The moment is M y ;
1
x x 2 dx ;
1
x 3 dx 15. A barrel initially contains 3 cups of sugar dissolved in 4 gallons of water. You then add
pure water at the rate of 2 gallons per minute while the mixture is draining out of a hole
in the bottom at 2 gallons per minute. Find the amount of sugar in the barrel after 1
minute.
Note: You must explicitly show the differential equation, the initial condition, the method of
solution, the general solution, the solution satisfying the initial condition and the final
answer.
Let SŸt be the cups of sugar in the barrel at time t.
So the initial condition is SŸ0 3: and the differential equation is:
dS 0cups • 2 gal " Scups • 2 gal " S
mn
mn
2
gal
4 gal
dt
IN
OUT
1
1
1
ln|S| " 12 t C
We solve: ; dS " ; dt
S oe C e " 2 t Ae " 2 t
|S| e " 2 tC
S
2
We use the initial condition: SŸ0 3 Ae 0 A
1
So the solution is: S 3e " 2 t
1
Finally after 1 minute there SŸ1 3e " 2 X 1.82cups of sugar in the barrel.