<< Chapter < Page Chapter >> Page >

Use a calculator to find e 0.5 . Round to five decimal places.

e 0.5 0.60653

Got questions? Get instant answers now!

Investigating continuous growth

So far we have worked with rational bases for exponential functions. For most real-world phenomena, however, e is used as the base for exponential functions. Exponential models that use e as the base are called continuous growth or decay models . We see these models in finance, computer science, and most of the sciences, such as physics, toxicology, and fluid dynamics.

The continuous growth/decay formula

For all real numbers t , and all positive numbers a and r , continuous growth or decay is represented by the formula

A ( t ) = a e r t

where

  • a is the initial value,
  • r is the continuous growth rate per unit time,
  • and t is the elapsed time.

If r > 0 , then the formula represents continuous growth. If r < 0 , then the formula represents continuous decay.

For business applications, the continuous growth formula is called the continuous compounding formula and takes the form

A ( t ) = P e r t

where

  • P is the principal or the initial invested,
  • r is the growth or interest rate per unit time,
  • and t is the period or term of the investment.

Given the initial value, rate of growth or decay, and time t , solve a continuous growth or decay function.

  1. Use the information in the problem to determine a , the initial value of the function.
  2. Use the information in the problem to determine the growth rate r .
    1. If the problem refers to continuous growth, then r > 0.
    2. If the problem refers to continuous decay, then r < 0.
  3. Use the information in the problem to determine the time t .
  4. Substitute the given information into the continuous growth formula and solve for A ( t ) .

Calculating continuous growth

A person invested $1,000 in an account earning a nominal 10% per year compounded continuously. How much was in the account at the end of one year?

Since the account is growing in value, this is a continuous compounding problem with growth rate r = 0.10. The initial investment was $1,000, so P = 1000. We use the continuous compounding formula to find the value after t = 1 year:

A ( t ) = P e r t Use the continuous compounding formula . = 1000 ( e ) 0.1 Substitute known values for  P ,   r ,  and  t . 1105.17 Use a calculator to approximate .

The account is worth $1,105.17 after one year.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

A person invests $100,000 at a nominal 12% interest per year compounded continuously. What will be the value of the investment in 30 years?

$3,659,823.44

Got questions? Get instant answers now!

Calculating continuous decay

Radon-222 decays at a continuous rate of 17.3% per day. How much will 100 mg of Radon-222 decay to in 3 days?

Since the substance is decaying, the rate, 17.3 % , is negative. So, r   =   0.173. The initial amount of radon-222 was 100 mg, so a = 100. We use the continuous decay formula to find the value after t = 3 days:

A ( t ) = a e r t Use the continuous growth formula . = 100 e 0.173 ( 3 ) Substitute known values for  a ,   r ,  and  t . 59.5115 Use a calculator to approximate .

So 59.5115 mg of radon-222 will remain.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Using the data in [link] , how much radon-222 will remain after one year?

3.77E-26 (This is calculator notation for the number written as 3.77 × 10 26 in scientific notation. While the output of an exponential function is never zero, this number is so close to zero that for all practical purposes we can accept zero as the answer.)

Got questions? Get instant answers now!

Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
what is titration
John Reply
what is physics
Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
what is inorganic
emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
what's motion
Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
hello friend how are you
Muhammad Reply
fine, how about you?
Mohammed
hi
Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
Reofrir Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
Practice Key Terms 4

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask