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Interest calculations can be quantified by mathematical formulas. Suppose that we are concerned with making an investment of P dollars at an annual rate of interest of i for n years. Here P denotes the present value of the investment, n represents the number of years the money is to be invested, and i is the interest rate per annum. Let us denote by F the future value of the investment at the end of n years. The value of F can be calculated via the formula

F = P ( 1 + i ) n size 12{F=P` \( 1+i \) rSup { size 8{n} } } {}

Let us illustrate the use of this formula by means of a problem.

Question: A couple has just had a baby. The couple wishes to make an investment in an account that will be used to fund college expenses. The couple visits an investment advisor and establishes a tax-free college savings account. By tax-free, we mean an account where income tax is not paid yearly. The couple is advised that this feature will enable the account to grow faster.

The couple deposits $25,000 in the account. The couple is told that the value of the account will appreciate at an annual rate of 8% . What will be the value of the account when the child turns 18 years of age?

Solution: We are asked to find the value of F . From the problem statement, P is $25,000, i is 0.08 and n is 18. We substitute into the interest formula to obtain the result

F = ( 25 , 000 ) ( 1 + 0 . 08 ) 18 = ( 25 , 000 ) ( 1 . 08 ) 18 = ( 25 , 000 ) ( 3 . 996 ) = 99 , 900 size 12{F= \( "25","000" \) ` \( 1+0 "." "08" \) rSup { size 8{"18"} } = \( "25","000" \) ` \( 1 "." "08" \) rSup { size 8{"18"} } = \( "25","000" \) ` \( 3 "." "996" \) ="99","900"} {}

We conclude that the account will be worth $99,900 at the end of 18 years.

Let us now a related problem that involves the use of logarithms.

Question: An individual inherits $10,000 from a relative. The individual wishes to invest the sum in a tax-free account. He/she plans to use the proceeds of this account as a down payment on a future purchase of a home. The annual interest rate of the account is 6%,

The individual anticipates that he/she will need at least $20,000 for the down payment. How long will it take for the value of the account to grow to $20,000?

Solution: Once again, we begin with the identification of the parameters of the problem. Here, P is 10,000, i is 0.06, and F is 20,000. We incorporate these values into the interest formula

20 , 000 = ( 10 , 000 ) ( 1 + 0 . 06 ) n size 12{"20","000"= \( "10","000" \) ` \( 1+0 "." "06" \) rSup { size 8{n} } } {}

Let us divide each side by (10,000) and re-arrange terms

( 1 . 06 ) n = 2 size 12{ \( 1 "." "06" \) rSup { size 8{n} } =2} {}

Now let us take the logarithm of each side of the equation. We will use 1.06 as the base of the logarithm

n = log 1 . 06 ( 2 ) size 12{n="log" rSub { size 8{1 "." "06"} } \( 2 \) } {}

The base 1.06 logarithm is related to the base 10 logarithm as follows

log 1 . 06 ( x ) = log 10 ( x ) log 10 ( 1 . 06 ) size 12{"log" rSub { size 8{1 "." "06"} } \( x \) = { {"log" rSub { size 8{"10"} } \( x \) } over {"log" rSub { size 8{"10"} } \( 1 "." "06" \) } } } {}

We will use this relationship to help us find the value for n

n = log 10 ( 2 ) log 10 ( 1 . 06 ) = 0 . 301 0 . 0253 = 11 . 90 size 12{n= { {"log" rSub { size 8{"10"} } \( 2 \) } over {"log" rSub { size 8{"10"} } \( 1 "." "06" \) } } = { {0 "." "301"} over {0 "." "0253"} } ="11" "." "90"} {}

So the individual should plan on waiting 11.90 years for the account to grow to a value of $20,000.

Problems such as the previous one often make use of the relationship

log a ( x ) = log b ( x ) log b ( a ) size 12{"log" rSub { size 8{a} } \( x \) = { {"log" rSub { size 8{b} } \( x \) } over {"log" rSub { size 8{b} } \( a \) } } } {}

This relationship allows one to convert from the logarithm of any base to a logarithm of another base. Typically, scientific calculators are only able to compute base 10 and natural (base e ) logarithms. One should become acquainted with the conversion of logarithms of other bases to base 10 or base e logarithms.

Exercises

  1. The power of the signal entering an amplifier is 15 mW. The power of the signal that leaves the amplifier is 25 W. Express the gain of the amplifier in decibels.
  2. Unlike an amplifier, some devices reduce the power of input signals. This process is called attenuation. Suppose that the power that enters an attenuator is 60 W and the power at the output is 0.9 W. Express the gain of the attenuator in decibels.
  3. Consider a signal processing scheme such as that shown in Figure 2. The power of the input signal before it passes through the noisy channel is 40 W. After passing through the noisy channel, the original signal is corrupted by noise. The noise component has a power of 10 W. What is the Signal to Noise ratio of the signal that emerges from the noisy channel?
  4. Consider the situation described in exercise 3. The noisy signal enters a signal processor. The signal processor diminishes the noise power of the signal by 5%, while diminishing the power in the noise component by 95%, What is the SNR of the output of the signal processor?
  5. Consider the circuit shown in Figure 3. Let us replace the resistor with another whose value is 200 kΩ. (a) What is the new value of the time constant of the circuit. (b) Find the value of the transient response when t = 0.8 seconds. (c) Determine the value of time at which the transient response decays to a value of 1 Volt.

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Good
Can someone give me problems that involes radical expressions like area,volume or motion of pendulum with solution
BUGAL Reply

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Source:  OpenStax, Math 1508 (laboratory) engineering applications of precalculus. OpenStax CNX. Aug 24, 2011 Download for free at http://cnx.org/content/col11337/1.3
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