MATLAB: An Introduction with Applications 4th Edition

1.10 Problems 27

When the script file is executed, the following (the values of the variables B, t,years, and months) is displayed in the Command Window:

1.10 PROBLEMS

The following problems can be solved by writing commands in the CommandWindow, or by writing a program in a script file and then executing the file.

1. Calculate:

(a) (b)

2. Calculate:

(a) (b)

3. Calculate:

(a) (b)

4. Calculate:

(a) (b)

>> format short gB = 20011

t = 16.374

years = 16

months = 5

The values of the variables B, t,years, and months are displayed(since a semicolon was not typed at theend of any of the commands that calcu-late the values).

14.82 6.52+( )3.82

---------------------------------- 552 14+

-------------------+ 3.5–( )3 e6

524ln-------------- 2061 3⁄+ +

16.52 8.4 70–( )

4.32 17.3–----------------------------------------- 5.23 6.42– 3+

1.68 2–----------------------------------- 13.3

5----------⎝ ⎠⎛ ⎞

1.5+

15 10 3.72+

10 1365( )log 1.9+-------------------------------------------⎝ ⎠⎛ ⎞

2.53 16 21622

---------–⎝ ⎠⎛ ⎞

1.74 14+------------------------------------ 20504+

2.32 1.7⋅

1 0.82–( )2 2 0.87–( )2+------------------------------------------------------------------- 2.34 1

2---2.7 5.92 2.42–( ) 9.8 51ln+ +

28 Chapter 1: Starting with MATLAB

5. Calculate:

(a) (b)

6. Define the variable x as x = 2.34, then evaluate:

(a) (b)

7. Define the variable t as t = 6.8, then evaluate:

(a) (b)

8. Define the variables x and y as x = 8.3 and y = 2.4, then evaluate:

(a) (b)

9. Define the variables a, b, c, and d as:

, , , and , then evaluate:

(a) (b)

10. A cube has a side of 18 cm.(a) Determine the radius of a sphere that has the same surface area as the

cube.(b) Determine the radius of a sphere that has the same volume as the cube.

11. The perimeter P of an ellipse with semi-minor axes a and

b is given approximately by: .

(a) Determine the perimeter of an ellipse with in.and in.

(b) An ellipse with has a perimeter of cm. Determine a and b.

12. Two trigonometric identities are given by:

(a) (b)

For each part, verify that the identity is correct by calculating the values of theleft and right sides of the equation, substituting .

7π9

------⎝ ⎠⎛ ⎞sin

57---π⎝ ⎠⎛ ⎞cos2

----------------------- 17--- 5

12------π⎝ ⎠⎛ ⎞tan+ 64°tan

14°cos2------------------- 3 80°sin

0.93--------------------– 55°cos

11°sin-----------------+

2x4 6x3– 14.8x2 9.1+ +e2x

14 x2 x–+------------------------------

t2 t3–( )ln 752t------ 0.8t 3–( )cos

x2 y2 x2

y2-----–+ xy x y+–

x y–x 2y–--------------⎝ ⎠

⎛ ⎞2 x

y--–+

a 13= b 4.2= c 4b( ) a⁄= d abca b c+ +---------------------=

a bc d+------------ d

c---a

b--- a b2–( ) c d+( )–+ a2 b2+

d c–( )--------------------- b a– c d–+( )ln+

a bP 2π 1

2--- a2 b2+( )=

a 9=b 3=

b 2a= P 20=

4xsin 4 x xcossin 8 xsin3 xcos–= 2xcos 1 xtan2–1 xtan2+----------------------=

x π9---=

1.10 Problems 29

13. Two trigonometric identities are given by:

(a) (b)

For each part, verify that the identity is correct by calculating the values of theleft and right sides of the equation, substituting .

14. Define two variables: alpha = 5π/8, and beta = π/8. Using these variables, showthat the following trigonometric identity is correct by calculating the values ofthe left and right sides of the equation.

15. Given: . Use MATLAB to calculate the following

definite integral: .

16. In the triangle shown cm, cm, andcm. Define a, b, and c as variables, and

then:(a) Calculate the angle α (in degrees) by substi-

tuting the variables in the Law of Cosines.(Law of Cosines: )

(b) Calculate the angles β and γ (in degrees)using the Law of Sines.

(c) Check that the sum of the angles is .

17. In the triangle shown in., in., and .Define a, b, and γ as variables, and then:(a) Calculate the length of c by substituting the variables in

the Law of Cosines.(Law of Cosines: )

(b) Calculate the angles α and β (in degrees) using the Lawof Sines.

(c) Verify the Law of Tangents by substituting the resultsfrom part (b) into the right and left sides of the equation.

(Law of Tangents:

4xtan 4 xtan 4 xtan3–1 6 xtan2– xtan4+--------------------------------------------= xsin3 1

4--- 3 xsin 3xsin–( )=

x 12°=

α βcossin 12--- α β–( )sin α β+( )sin+[ ]=

ax( ) xdcos2∫ 12---x 2axsin

4a-----------------–=

0.5x( ) xdcos2π9---

3π5

------

∫

A

BCa

bc α

βγ

a 9= b 18=c 25=

c2 a2 b2 2ab γcos–+=

180°

α

β

γ

a

b

c

A

B

Ca 5= b 7= γ 25°=


a b–a b+------------

12--- α β–( )tan

12--- α β+( )tan

-----------------------------------=


18. For the triangle shown, mm, mm,and mm. Define a, b, and c as variables,and then:(a) Calculate the angle γ (in degrees) by substituting

the variables in the Law of Cosines. (Law of Cosines: )

(b) Calculate the radius r of the circle inscribed in

the triangle using the formula .(c) Calculate the radius r of the circle inscribed in the triangle using the for-

mula , where .

19. In the right triangle shown cm and cm.Define a and c as variables, and then:(a) Using the Pythagorean Theorem, calculate b by

typing one line in the Command Window.(b) Using b from part (a) and the acosd function,

calculate the angle α in degrees by typing one linein the Command Window.

20. The distance d from a point to a plane is given by:

Determine the distance of the point from the plane. First define the variables A, B, C, D, x0, y0, and z0,

and then calculate d. (Use the abs and sqrt functions.)

21. The arc length s of the parabolic segment BOC is given by:

Calculate the arc length of a parabola with in.and in.

22. Oranges are packed such that 52 are placed in each box. Determine how manyboxes are needed to pack 4,000 oranges. Use MATLAB built-in functionceil.

α

β

γ a

b

c

r

a 200= b 250=c 300=


r 12--- a b c–+( ) 1

2---γ⎝ ⎠⎛ ⎞tan=

r s s a–( ) s b–( ) s c–( )s

-------------------------------------------------------= s 12--- a b c+ +( )=

a

b

c

α

a 16= c 50=

x0 y0 z0, ,( ) Ax By Cz D+ + + 0=

dAx0 By0 Cz0 D+ + +

A2 B2 C2+ +-----------------------------------------------------=

8 3 10–, ,( )2x 23y 13z 24–+ + 0=

b

aB

O

C

s 12--- b2 16a2+

b2

8a------ 4a b2 16a2++

b----------------------------------------⎝ ⎠⎛ ⎞ln+=

a 12=b 8=

1.10 Problems 31

23. The voltage difference between points aand b in the Wheatstone bridge circuit is:

Calculate the voltage difference when volts, ohms, ohms,

ohms, and ohms.

24. The prices of an oak tree and a pine tree are $54.95 and $39.95, respectively.Assign the prices to variables named oak and pine, change the display formatto bank, and calculate the following by typing one command: (a) The total cost of 16 oak trees and 20 pine trees.(b) The same as part (a), and add 6.25% sale tax.(c) The same as part (b) and round the total cost to the nearest dollar.

25. The resonant frequency f (in Hz) for the circuitshown is given by:

Calculate the resonant frequency whenhenrys, ohms, ohms,

and farads.

26. The number of combinations of taking r objects out of n objects is givenby:

A deck of poker cards has 52 different cards. Determine how many differentcombinations are possible for selecting 5 cards from the deck. (Use the built-in function factorial.)

27. The formula for changing the base of a logarithm is:

(a) Use MATLAB’s function log(x) to calculate . (b) Use MATLAB’s function log10(x) to calculate .

+V

R1 R3

R4R2

a b

Vab

Vab VR2

R1 R2+------------------

R4

R3 R4+------------------–⎝ ⎠

⎛ ⎞=

V 12=R1 120= R2 100=

R3 220= R4 120=

V

R1 R2

L C

f 12π------ LC

R12C L–

R22C L–

--------------------=

L 0.2= R1 1500= R2 1500=

C 2 10 6–×=

Cn r,

Cn r,n!

r! n r–( )!----------------------=

aNlog bNlog

balog---------------=

4 0.085log

61500log


28. The current I (in amps) t seconds after closing theswitch in the circuit shown is:

Given volts, ohms, andhenrys, calculate the current 0.003 seconds

after the switch is closed.

29. Radioactive decay of carbon-14 is used for estimating the age of organicmaterial. The decay is modeled with the exponential function ,where t is time, is the amount of material at , is the amount ofmaterial at time t, and k is a constant. Carbon-14 has a half-life of approxi-mately 5,730 years. A sample of paper taken from the Dead Sea Scrolls showsthat 78.8% of the initial ( ) carbon-14 is present. Determine the esti-mated age of the scrolls. Solve the problem by writing a program in a scriptfile. The program first determines the constant k, then calculates t for

, and finally rounds the answer to the nearest year.

30. Fractions can be added by using the smallest common denominator. Forexample, the smallest common denominator of 1/4 and 1/10 is 20. Use theMATLAB Help Window to find a MATLAB built-in function that determinesthe least common multiple of two numbers. Then use the function to showthat the least common multiple of: (a) 6 and 26 is 78. (b) 6 and 34 is 102.

31. The Moment Magnitude Scale (MMS), denoted , which is used to mea-sure the size of an earthquake, is given by:

where is the magnitude of the seismic moment in dyne-cm (measure ofthe energy released during an earthquake). Determine how many times moreenergy was released from the earthquake in Sumatra, Indonesia ( ),in 2007 than the earthquake in San Francisco, California ( ), in 1906.

32. According to special relativity, a rod of length L moving at velocity v willshorten by an amount , given by:

where c is the speed of light (about m/s). Calculate how much a rod2 meter long will contract when traveling at 5,000 m/s.

V +

L

RI V

R--- 1 e R L⁄( )t––( )=

V 120= R 240=

L 0.5=

f t( ) f 0( )ekt=f 0( ) t 0= f t( )

t 0=

f t( ) 0.788f 0( )=

MW

MW23---

10 M0log 10.7–=

M0

MW 8.5=

MW 7.9=

δ

δ L 1 1 v2

c2-----––⎝ ⎠

⎛ ⎞=

300 106×

1.10 Problems 33

33. The monthly payment M of a loan amount P for y years and interest rate r canbe calculated by the formula:

(a) Calculate the monthly payment of a $85,000 loan for 15 years and interestrate of 5.75% ( ). Define the variables P, r, and y and use themto calculate M.

(b) Calculate the total amount needed for paying back the loan.

34. The balance B of a savings account after t years when a principal P is investedat an annual interest rate r and the interest is compounded yearly is given by

. If the interest is compounded continuously, the balance isgiven by . An amount of $40,000 is invested for 20 years in anaccount that pays 5.5% interest and the interest is compounded yearly. UseMATLAB to determine how many fewer days it will take to earn the same ifthe money is invested in an account where the interest is compounded contin-uously.

35. The temperature dependence of vapor pressure p can be estimated by theAnteing equation:

where ln is the natural logarithm, p is in mm Hg, T is in kelvins, and A, B, andC are material constants. For toluene (C6H5CH3) in the temperature rangefrom 280 to 410 K the material constants are , , and

. Calculate the vapor pressure of toluene at 315 and 405 K.

36. Sound level in units of decibels (dB) is determined by:

where p is the sound pressure of the sound, and Pa is a refer-ence sound pressure (the sound pressure when dB). (a) The sound pressure of a passing car is Pa. Determine its sound

level in decibels. (b) The sound level of a jet engine is 110 decibels. By how many times is the

sound pressure of the jet engine larger (louder) than the sound of the pass-ing car?

M P r 12⁄( )1 1 r 12⁄+( ) 12y––--------------------------------------------=

r 0.0575=

B P 1 r+( )t=B Pert=

p( )ln A BC T+-------------–=

A 16.0137= B 3096.52=C 53.67–=

LP

LP 20 10pp

0-----⎝ ⎠

⎛ ⎞log=

p0

20 10 6–×=

LP 0=

80 10 2–×


37. Use the Help Window to find a display format that displays the output as aratio of integers. For example, the number 3.125 will be displayed as 25/8.Change the display to this format and execute the following operations:(a) (b)

38. The steady-state heat conduction q from a cylin-drical solid wall is determined by:

where k is the thermal conductivity. Calculate qfor a copper tube ( Watts/oC/m) of length

cm with an outer radius of cmand an inner radius of cm. The external temperature is C andthe internal temperature is C.

39. Stirling's approximation for large factorials is given by:

Use the formula for calculating 20!. Compare the result with the true valueobtained with MATLAB’s built-in function factorial by calculating theerror ( ).

40. A projectile is launched at an angle andspeed of . The projectile’s travel time ,maximum travel distance , and maximumheight are given by:

, ,

Consider the case where ft/s and . Define and as

MATLAB variables and calculate , , and ( ft/s2).

5 8⁄ 16 6⁄+ 1 3⁄ 11 13⁄– 2.72+

T2

r1T1r2

Lq 2πLk

T1 T2–

r2

r1----⎝ ⎠⎛ ⎞ln

-----------------=

k 401=L 300= r2 5=

r1 3= T2 20°=

T1 100°=

n! 2πn ne---⎝ ⎠⎛ ⎞

n=

Error TrueVal ApproxVal–( ) ⁄= TrueVal

V0

θο x

y

xmax

hmax

θV0 ttravel

xmax

hmax

ttravel 2V0

g----- θ0sin= xmax 2

V02

g------ θ0 θ0cossin=

hmax 2V0

2

g------ θ0sin2=

V0 600= θ 54°= V0 θ

ttravel xmax hmax g 32.2=

MATLAB: An Introduction with Applications 4th Edition

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