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An expression for the area of a circle

From your earlier coursework, you should know that the area of a circle is given by

A = PI * r^2

where

  • A is the area.
  • PI is the mathematical constant 3.14159...
  • r is the radius of the circle.

Proportional to the square of the radius

From this, we can conclude that the area of a circle is not proportional to the radius. Instead, it is proportional to the square of the radius as in

A $ r^2

If you change the radius...

If you change the value of the radius, the area changes in proportion to the square of the radius. If the radius doubles, the area increases by four. If theradius is decreased by 25-percent, the area decreases by more than 25-percent. This is illustrated by the script in Listing 3 .

Listing 3 . Area is proportional to radius squared.
<!-- File JavaScript03.html --><html><body><script language="JavaScript1.3">var r = 10 var A = Math.PI * Math.pow(r,2)document.write("r =" + r + ", A = " + A + "<br/>") //Multiply r by 2. Then display r and Cr = r * 2 var A = Math.PI * Math.pow(r,2)document.write("r =" + r + ", A = " + A + "<br/>")//Reduce r by 25%, Then display r and C r = r * (1 - 25/100)var A = Math.PI * Math.pow(r,2) document.write("r =" + r +", A = " + A + "<br/>")//Compute and the display the cube root // of a number.var X = Math.pow(8,1/3) document.write("Cube root of 8 = " + X)</script></body></html>

The JavaScript Math.pow method

Listing 3 calls a built-in JavaScript method that I have not used before: Math.pow . This method is called to raise a value to a power. It requires two parameters. The first parameter is the valuethat is to be raised to a power and the second parameter is the power to which the value is to be raised.

The method returns the value raised to the power.

Fractional exponents

Although this topic is not directly related to the discussion on proportionality, as long as I am introducing the method named Math.pow , I will point out the it is legal for the exponent to be a fraction. The last little bit of code in Listing 3 raises the value 8 to the 1/3 power. This actually computes the cube root of the value 8. As you shouldbe able to confirm in your head, the cube root of 8 is 2, because two raised to the third power is 8.

Output from the script

When you open the script shown in Listing 3 in your browser, the text shown in Figure 3 should appear in your browser window.

Figure 3 . Screen output for Listing #3.
r =10, A = 314.1592653589793 r =20, A = 1256.6370614359173r =15, A = 706.8583470577034 Cube root of 8 = 2

An examination of the first three lines of text in Figure 3 should confirm that they satisfy the proportionality rules for the square of theradius described earlier .

The last line of text in Figure 3 confirms that the Math.pow method can be used to compute roots by specifying fractional exponents as thesecond parameter.

Run the scripts

I encourage you to run the scripts that I have presented in this lesson to confirm that you get the same results. Copy the code for each script into atext file with an extension of html. Then open that file in your browser. Experiment with the code, making changes, and observing the results of your changes. Makecertain that you can explain why your changes behave as they do.

Miscellaneous

This section contains a variety of miscellaneous information.

Housekeeping material
  • Module name: GAME 2302-0330 Scale Factors, Ratios, and Proportions
  • File: Game0330.htm
  • Revised: 01/25/16
Disclaimers:

Financial : Although the Connexions site makes it possible for you to download a PDFfile for this module at no charge, and also makes it possible for you to purchase a pre-printed version of the PDF file, you should be aware thatsome of the HTML elements in this module may not translate well into PDF.

I also want you to know that, I receive no financial compensation from the Connexions website even if you purchase the PDF version of the module.

In the past, unknown individuals have copied my modules from cnx.org, converted them to Kindle books, and placed them for sale on Amazon.comshowing me as the author. I neither receive compensation for those sales nor do I know who does receive compensation. If you purchase such a book, pleasebe aware that it is a copy of a module that is freely available on cnx.org and that it was made and published without my prior knowledge.

Affiliation : I am a professor of Computer Information Technology at Austin Community College in Austin, TX.

-end-

Questions & Answers

What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
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Source:  OpenStax, Game 2302 - mathematical applications for game development. OpenStax CNX. Jan 09, 2016 Download for free at https://legacy.cnx.org/content/col11450/1.33
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