3.

9 Problems 87

(x+
3. For the function y =
(x+ 1) x
7j , calculate the value of y for the following

values of x using element-by-element operations: 1.5, 2.5, 3.5, 4.5, 5.5, 6.6 .

· 2 sinx+ cos2x .
4. For the funct10n y = . , calculate the value of y for the followmg
2
sm x

values ofxusing element-by-element operations: 20°, 30°, 40°, 50°, 60°, 70°.

5. The radius, r, of a sphere can be calculated from its surface area, s, by:

--
Ji1ic
r= 2
The volume, V, is given by:
41tr3
V= -

3
Determine the volume of spheres with surface area of 50, 100, 150, 200, 250,
and 300 tt2. Display the results in a two-column table where the values of s
and V are displayed in the first and second columns, respectively.

6. The electric field intensity, E(z), due to a ring z

of radius R at any point z along the axis of the
ring is given by:
- ')... Rz
E(z)
2Eo(z2 + R2)312
where J... is the charge density,
e0 = 8.85 x I0-12 is the electric constant, and
R is the radius of the ring. Consider the case
where J... = 1.7 x to-7 C/m and R = 6 em.
(a) Determine E(z) at z = 0, 2, 4, 6, 8, and 10 em.
(b) Determine the distance z where E is maximum. Do it by creating a vector
z with elements ranging from 2 em to 6 em and spacing of 0.01 em. Cal­
culate E for each value of z and then fmd the maximum E and associated z
with MATLAB's built-in function max.

7. The voltage Vc(t) (in V) and the current i(t)

(in Amp) t seconds after closing the switch in
the circuit shown are given by:
Vc(t) = V0(1- e-tl�o )
v.
i(t) = _Qe-tl'to
R
where 'to = RC is the time constant. Consider the case where V0 = 24 V,
R = 3800 Q and C = 4000 x I0-6 F. Determine the voltage and the current
during the first 20 s after the switch is closed. Create a vector with values of
88 Chapter 3: Mathematical Operations with Arrays

times from 0 to20 s with spacing of2 s, and use it for calculating Vc(t) and
i(t). Display the results in a three-column table where the values of time,
voltage and current are displayed in the first, second, and third columns,
respectively.

8. The length lui (magnitude) of a vector u = xi+yj+zk is given by

u
l i = Jx2+y2+z2. Given the vector u = 23.5i- 17j+ 6k, determine its
length in the following two ways:
(a) Define the vector in MATLAB, and then write a mathematical expression
that uses the components of the vector.
(b) Define the vector in MATLAB, then determine the length by writing one
command that uses element-by-element operation and MATLAB built-in
functions sum and sqrt.

9. A vector wL of length Lin the direction of a vector u = xi+yj+zk can deter­

mined by wL = Lun (multiplying a unit vector in the direction ofu by L ).
The unit vector un in the direction of the vector u is given by
xi+yj+zk . . .
un = ll..rA'TLAB command, determme a vector
. By wntmg one 1v.lr\.
Jx2+y2+z2
of length 1 8 in the direction of the vector u = 7i- 4j- 11k .

1 0. The following two vectors are defmed in MATLAB:

v = [15,8,-6] u = [3,-2,6]
By hand (pencil and paper) write what will be displayed if the following com­
mands are executed by MATLAB. Check your answers by executing the com­
mands with MATLAB.
(a) v. /u (b) u' *v (c) u*v'

11. Two vectors are given:

u = 5i- 6j+9k and v = 11i+7j- 4k

Use MATLAB to calculate the dot product u · v of the vectors in three ways:

(a) Write an expression using element-by-element calculation and the MAT-

LAB built-in function sum.
(b) Define u as a row vector and vas a column vector, and then use matrix
multiplication.
(c) Use the MATLAB built-in function dot.

12. Defme the vector v = [2 3 4 5 6]. Then use the vector in a mathematical
expression to create the following vectors:
(a) a = [4 6 8 10 12] (b) b = [8 27 64 125 216]
(c) c = [22 33 44 5s 66] (d) d = [1 1.5 2 2.5 3]
3.9 Problems 89

13. Define the vector v = [8 6 4 2] . Then use the vector in a mathematical

expression to create the following vectors:

(a) a = [1 1 1 1] (b) b =
[ 1 1 1 1 ]
82 62 42 22
(c) c =
[J8 J6 J4 J2]
1 1 1 1
(d) d = [3 1 -1 -3 ]

14. Define x andy as the vectors x= [1, 2, 3, 4, 5] and y= [2, 4, 6, 8, 10]. Then
use them in the following expressions to calculate z using element-by-element
calculations.

(a) z =
(x +y)2
x-y
(b) w = xln(x2 + y2) + J(y-x)2 Y3

15. Define r and s as scalars r = 1 .6 x 103 and s = 14. 2 , and, t, x, andy as vec­
tors t= [1, 2, 3, 4, 5] , x= [0, 2, 4, 6, 8], and y= [3, 6, 9, 12, 15] . Then use
these variables to calculate the following expressions using element-by-ele­
ment calculations for the vectors.

(a) G =
r
xt+ -(y2-x)t
s2
(b) R =
r
(- x�;ytz -s2(y-0.5x2)t
)

16. The area of a triangle ABC can be calculated by z

c (-5, 2,11)
lrAB x rAd/2, where rAB and rAe are vectors
-

connecting the vertices A and B and A and C,

respectively. Determine the area of the triangle
shown in the figure. Use the following steps in
a script file to calculate the area. First, defme x

the vectors r oA, rOB and rae from knowing y

A (8, 5,-4)
the coordinates of points A, B, and C. Then
determine the vectors r AB and rAe from r oA, rOB and rae. Finally, determine
the area by using MATLAB's built-in functions cross, sum. and sqrt.

17. The volume of the parallelepiped shown can be

calculated by r0B (r0A x rAe). Use the follow­
·

ing steps in a script file to calculate the area.

Defme the vectors r oA , rAe, and rOB from
knowing position of points A, B, and C.
Determine the volume by using MATLAB 's
built-in functions dot and cross.
90 Chapter3: M&thematicalOperationswidlArrays

18. Define the vectors:

u = Sl-2j +4k, v = -21+ 7J +3k, and w = 81 + lj -3k
Use the vectors to verify the identity:
(u+v)·[(v+w)x(w+u)] = 2u·(vxw)
Use MATLAB's built-in :functions cross and dot, calculate the value of the
leftand right sides of the identity.

19. The dot product can be used for determining the

angle between two vecton:
z

e = cos
-
• --"
(lr1• r� ..

r
r.ll :z
Use MATLAB's built-in functions acosd,
sqrt, and dot to find the angle (in degrees)
between r1 = 6i-3j +2k and
r 2
= 21+9J + lOk.
�that lrl = Jr:i .

20 Use MAILAB to show that the angle inscribed y

in a semi-circle is a right angle. Use the follow­

ing steps in a script file to calculate the angle.
Define a variable with the value of the x coordi­
nate of point A. Determine the y coordinate of
point A using the equation x2+y:z = R2. Define
vectors that con-espond to the position of points A, B, and C and use them for
determining position vectorsr.4B r..cc.
and Calculate theangle ex in two ways.

=
·
- and th en by using the equa-
b using
First y a
(:::;;� ,
the equation
cos 1 1 11
r r
tion a sm-•(l ..cB ..td\.
= X
lr.ullr..cd) Both should give 90°.

21. The position as a function of time (x(t),y(t)) y

of a projectile fired with a speed of v0 at an '
angle a is given by '
'

= =
\
x(t) v0cosa t y(t) v0sina. t- !gt2 \
2
· ·

= 9.81 m/s2• The polar coordinates of

\

where g \
\ X
the projectile at time t are (r(t), 8(1)), where

r(t) = ,/x(t)2+y(t)2 and tan8(t) = ;�:� . Consider the case where

v0 = 162m/sand ex = 70°. Determine r(t) and 8(t) for t = 1, 6, 11, . .. , 3ls.

3.9 Problems 91

22. Use MATLAB to show that the sum of the infinite series I 2n.� converges to
n=O
e
2
• Do this by computing the sum for:
(a) n = 5 , (b) n = 10, (c) n = 50
For each part create a vector n in which the first element is 0, the increment is
1 and the last term is 5, 10, or 50. Then use element-by-element calculations to

�
create a vector in which the elements are 2 . Finally, use MATLAB's built-in
n.
function sum to sum the series. Compare the values to e
2
(use format
long to display the numbers).

23. Use MATLAB to show that the sum of the infmite series I <91n10)n con-
n= I
verges to 1n 10 . Do this by computing the sum for
(a) n = 10, (b) n = 50, (c) n = 100
For each part, create a vector n in which the frrst element is 1, the increment is
1 and the last term is 10, 50 or 100. Then use element-by-element calculations
(9/10 n
to create a vector in which the elements are ) . Finally, use MAT-
n

LAB's built-in function sum to sum the series. Compare the values to ln10
(use format long to display the numbers).

24. According to Zeno's paradox any object in motion must arrive at the halfway
point before it can arrive at its destination. Once arriving at the halfway point,
the remaining distance is once again divided in half and so on to infmity.
Since it is impossible to complete this process, Zeno concluded all motion
must be an illusion. Letting the length be unity, Zeno's paradox can be written

in terms of the infinite sum I .!.n = 1 . To see how quickly this series con-
n= I2
verges to 1, compute the sum for:
(a) n= 5, (b) n = 10, (c) n= 40
For each part create a vector n in which the first element is 1, the increment is
1, and the last term is 5 , 10, or 40. Then use element-by-element calculations to

create a vector in which the elements are .!. . Finally, use the MATLAB built-
n
2
in function sum to add the terms of the series. Compare the values obtained in
parts (a), (b), and (c) with the value of 1.
92 Chapter 3: Mathematical Operations with Arrays

cos(2x)-1 .
25. Show that lim =
4
x-70 cosx-1
Do this by first creating a vector x that has the elements 1.0, 0.5, 0.1, 0.01,
0.001, and 0.0001. Then, create a new vectoryin which each element is deter-
cos(2x)
mined from the elements of x by
cosx-
� 1 . Compare the elements of y
with the value 4 (use format long to display the numbers).

. x113-1 4
26. Show that bm .
x-7Ixll4_1
=
-- -

3
Do this by first creating a vector x that has the elements 2.0, 1.5, 1.1, 1.01,
1.001, 1.00001, and 1.0000001. Then, create a new vector yin which each ele­
x113-1
ment is determined from the elements of x by � . Compare the elements
X -1
ofywith the value 4/3 (use format long to display the numbers).

27. The demand for water during a fire is often the most important factor in the
design of distribution storage tanks and pumps. For communities with popula­
tions less than 200,000, the demand Q (in gallons/min) can be calculated by:
Q 1020Jil(l-0.01Jil)
=

where P is the population in thousands. Set up a vector for P that starts at 10

and increments by 10 up to 200. Use element-by -element computations to
determine the demand Q for each population in P.

28. The ideal gas equation states that P =

�
n T
, where P is the pressure, V is the
volume, T is the temperature, R 0.08206 (L atm)/(mol K) is the gas con­
=

stant, and n is the number of moles. Real gases, especially at high pressure,
deviate from this behavior. Their response can be modeled with the van der
Waals equation
nRT _n2a
p
V-nb
=

J12
where a and b are material constants. Consider 1
1 ) of nitrogen mole ( n =

gas at T 300K. (For nitrogen gas a

=
2 2
1.39 (L atm)/mol , and b
= 0.0391 =

L/mol.) Create a vector with values of Vs for 0.1� V� 1 L, using increments

of 0.02 L. Using this vector calculate P twice for each value of V, once using
the ideal gas equation and once with the van der Waals equation. Using the
p
two sets of values for P, calculate the percent of error ( ide� -Pwaals 100) for
waals
each value of V. Finally, by using MATLAB's built-in function max, deter­
mine the maximum error and the corresponding volume.
3.9 Problems 93

29. Create the following three matrices:

1
[ � -3 5 [0 :-2 1 -3 4
[0 ] -1
A= 2 2 4 B= 5 1 -6 c= 8 2
-20 6 2 7 -1 -3 5 3
(a) Calculate A+ B and B +A to show that addition of matrices is commuta­
tive.

(b) Calculate A+(B+C) and (A+ B)+C to show that addition of matrices is
associative.

(c) Calculate 3(A+C) and 3A + 5C to show that, when matrices are multi­
plied by a scalar, the multiplication is distributive.

(d) Calculate A*(B+C) and A* B+ A*C to show that matrix multiplication is

distributive.

30. Use the matrices A, B, and C from the previous problem to answer the follow­
ing:

(a) Does A*B= B*A? (b) Does A*(B*C) = (A*B)*C?

31. Create a 4 x 4 matrix A having random integer values between 1 and 10. Call
the matrix A and, using MATLAB, perform the following operations. For each
part explain the operation.

(a) A* A (b) A.*A (c) A\A

(d) A. I A (e) det (A) (e) inv (A)

32. The magic square is an arrangement of numbers in a square grid in such a way
that the sum of the numbers in each row, and in each column, and in each
diagonal is the same. MATLAB has a built-in function magic (n) that
returns an n x n magic square. In a script file create a (6 x 6) magic square,
and then test the properties of the resulting matrix by fmding the sum of the
elements in each row, in each column and in both diagonals. In each case, use
MATLAB's built-in function sum. (Other functions that can be useful are
diag and fliplr.)

33. Solve the following system of three linear equations:

- 4x+3y+z= -18.2
5x+6y- 2z = -48.8
2x- 5y+4.5z = 92.5
94 Chapter 3: Mathematical Operations with Arrays

34. Solve the following system of five linear equations:

2.5a-b+3c+ 1.5d-2e = 57. 1

3a+4b-2c+2.5d-e = 27.6
-4a + 3b + c-6d+2e = -81.2
2a+3b+c-2.5d+4e = -22.2
a+2b+5c-3d+4e = -12.2

35. A food company manufactures five types of 8 oz Trail mix packages using
different mixtures of peanuts, almonds, walnuts, raisins, and M&Ms. The
mixtures have the following compositions:

Peanuts(oz) Almonds(oz) Walnuts(oz) Raisins(oz) M&Ms(oz)

Mix 1 3 1 1 2 1
Mix2 1 2 1 3 1
Mix3 1 1 0 3 3
Mix4 2 0 3 1 2
Mix5 1 2 3 0 2

How many packages of each mix can be manufactured if 128 lb of peanuts,

118 lb of almonds, 112 lb of walnuts, 112 lb of raisins, and 104 lb of M&Ms
are available? Write a system of linear equations and solve.

36. The electrical circuit shown consists of resis­

tors and voltage sources. Determine i1, i2, i3 ,

and i4 , using the mesh current method based

on Kirchhoff's voltage law (see Sample
Problem 3-4).

37. The electrical circuit shown consists of

resistors and voltage sources. Determine
i1, i2, i3, i4 and i5 , using the mesh current
method based on Kirchhoff's voltage law
(see Sample Problem 3-4).

v, = 40V, V2 = 30V, V3 = 36V,

R1 = 160, R2 = 200, R3 = 100
R4 = 140, R5 = 8 Q, R6 = 16 Q,
R1 = 10 Q, R8 = 15 0, R9 = 6 0,
R 10 = 4 Q.