Exponential Growth

Key Questions

• How do you graph exponential growth?

  For graphing an exponential function, basically three things have to be observed. First is the end behavior of the function. One should be able to figure out what happens when x goes to +infy and what happens when x goes to - infy. Next is the point at which the graph would crosss y -axis, that is when x=0. To sketch the general out line, a few points can by worked out by assigning different values to x. which can be plotted.

  bp · 1 · Apr 6 2015
• What is the exponential growth formula?

  Answer:

  A function, whose derivative is proportional to itself, is an exponential.

  Explanation:

  If a function has a derivative proportional to itself, and the proportionality factor is real, then the function grows or decays exponentially.

  t0hierry · 1 · Jun 30 2018
• What is Exponential Growth?

  Answer:

  #y# is an exponential growth function of #x# if #y=a*b^{x}# for some #a>0# and some #b>1#. This is often written as #y=a*e^{k*x}#, where #e=2.718281828459...# and #e^{k}=b# (so #k=ln(b)>0#).

  Explanation:

  This is the definition of what an exponential growth function is.

  On the other hand, if #0 < b < 1#, then the function is called an exponential decay function (and #k=ln(b) < 0#).

  You should graph examples of functions like these on your calculator to see what their graphs look like.

  The number #e=2.718281828459...# is a "special" number in mathematics (special like #pi# is special). When it's used as the base of an exponential function in calculus, the resulting calculations you can do with it are simpler than they would be if you used some other base.

  The function #ln(x)=log_{e}(x)# is called the "natural logarithm" and the same comments apply to it.

  Bill K. · 2 · Jul 22 2015

• What is Exponential Growth?
• What is the exponential growth formula?
• How does exponential growth differ from linear growth?
• How do you graph exponential growth?
• What is the y-intercept of all exponential growth functions?
• How do you graph #y=\frac{1}{2} \cdot 4^x#?
• How do you graph #f(x)=2 \cdot 3^x#?
• How much money will Nadia have in the bank by her 21st birthday if she received $200 for her 10th birthday and saved it in a bank with a 7.5% interest rate compounded yearly?
• How do you graph #f(x)=2*3^x#?
• How do you determine the multiplier for exponential growth and decay?
• How do we use exponential growth and decay in real life?
• How do you solve exponential growth and decay problems?
• What do exponential growth and decay have in common?
• How do you identify equations as exponential growth or exponential decay?
• Is the equation #V(T)=100(1.08)^T# an exponential growth or decay?
• Without graphing, how do you determine whether each equation #Y=72(1.6)^x# represents exponential growth of exponential decay?
• The equation y=6.72(1.014)^x models the world population y, in billions of people, x-years after the year 2000. Find the year in which the world population is about 10 billion?
• Under ideal conditions, a population of rabbits has an exponential growth rate of 11.5% per day. Consider an initial population of 900 rabbits, how do you find the growth function?
• In 1992, the city of Chicago had 6.5 million people. In 2000 they project Chicago will have 6.6 million people. If Chicago's population grows exponentially, how many people will live in Chicago in 2005?
• The initial population is 250 bacteria, and the population after 9 hours is double the population after 1 hour. How many bacteria will there be after 5 hours?
• You invest your birthday money in a savings account for at 6.5% interest compounded annually. How do you determine how much birthday money you need to invest to have $350 after 5 years?
• The Marriot family bought a new apartment three years ago for $65,000. The apartment is now worth $86,515. Assuming a steady rate of growth, what was the yearly rate of appreciation?
• You deposited $2000 into a college savings account 5 years ago. if the account pays 2.5% annual interest, compounded quarterly, how do you find the current balance of the savings account? please show work thank you ! (=?
• At the time the authorities estimated the population to be 3000 frogs. Now five years after first being introduced to this area the population is estimated to be 196000 frogs. When will the population reach 2 000 000 frogs?
• A sample of 2600 bacteria selected from this population reached the size of 2817 bacteria in two hours. How do you find the continuous growth rate per hour?
• A bacteria culture doubles every 0.25 hours. At time 1.25 hours, there are 40 000 bacteria present. How many bacteria were present initially?
• Question #b07ae
• Is the function #f(x) = (2/3)^x# increasing or decreasing?
• Is the function #f(x) = (5/2)^x# increasing or decreasing?
• Is the function #f(x) = (18)^x# increasing or decreasing?
• Is the function #f(x) = 4^-x# increasing or decreasing?
• Is the function #f(x) = (1.25)^x# increasing or decreasing?
• Is the function #f(x) = (1/5)^x# increasing or decreasing?
• Suppose one of your ancestors invested $500 in 1800 in an account paying 4% interest compounded annually. How do you write an exponential function to model the situation?
• How do you find the formula of an exponential graph given A = (1, 12) and B = (2, 48)?
• How do you find the formula of an exponential graph given #f(2)=9/4# and #f(-2)=4/9#?
• How do you find the formula of an exponential graph given (0, -2) and (2, -50)?
• How do you find the formula of an exponential graph given (0,4) and (1,4.3)?
• When Brie was born, she received a gift $500 from her grandparents. Her parents put it into an account at an annual yield of 5.2%. Brie is now 12 years old. How much is this gift worth now?
• If I invest $10 for example, and I make 1% extra per day, what would the equation be?
• How many hours will it take a culture of bacteria to increase from 20 to 2,000 if the growth rate per hour is 85%?
• A piece of machinery valued at $250,000 depreciates at a fixed rate of 12% per year. After how many years will the value have depreciated to $100,000?
• The population of rabbits in an area is modeled by the growth equation P(t)=8e^0.26t, where P is in thousands and t is in years. How long will it take for the population to reach 25,000?
• Suppose that the wealth of a business owner is increasing exponentially. In 1993, he had $40 million. In 2001, he had $55 million. How much money will he have in 2010?
• How do you determine if the equation #y=0.25(1.03)^5# represents exponential growth or decay?
• How do you determine if the equation #y = 2(0.45)^x# represents exponential growth or decay?
• In 1990, the population of the city of Orange was 110,658 and grew to 136,392 in 2008. Assuming exponential growth, what would the population of Orange be in 2030?
• The half-life of Radium-226 is 1590 years. If a sample contains 100 mg, how many mg will remain after 4000 years?
• What is the exponential growth given A=1,500,000, r= 5.5%, n= 7?
• A painting, purchased for $10 000 in 1990, increased in value by 8% per year. How do you find the value of the painting in the year 2000?
• If you invest $2000 in a bank offering 10% interest compounded weekly, how do you find the value of your investment after five years?
• How do you determine if the equation #y = (0.25)^x# represents exponential growth or decay?
• How do you determine if the equation #y = (7)^x# represents exponential growth or decay?
• How do you determine if the equation #y = 1/3 (0.1)^x# represents exponential growth or decay?
• How do you determine if the equation #y = (2/3)^x# represents exponential growth or decay?
• A country's population in 1995 was 164 million. In 2001 it was 169 million. How do you estimate the population in 2015 using exponential growth formula?
• A curve passes through the point (0,5) and has the property that the slope of the curve at every point P is twice the y-coordinate of P. What is the equation of the curve?
• A colony of bacteria is grown under ideal conditions in a lab so that the population increases exponentially with time. At the end of 3 hours there are 10,000 bacteria. at the end of 5 hours there are 40000. How many bacteria were present initially?
• A country's population in 1994 was 195 million. In 2002 it was 199 million. How do you estimate the population in 2016 using the exponential growth formula?
• How do you determine if the equation #f(x) = 4(.07)^x# represents exponential growth or decay?
• How do you determine if the equation #f(x) = 7(1/4)^x# represents exponential growth or decay?
• How do you determine if the equation #f(x) = 2(.44)^x# represents exponential growth or decay?
• How do you determine if the equation #f(x) = 6^x# represents exponential growth or decay?
• How do you determine if the equation #f(x) = 3(.11)^x# represents exponential growth or decay?
• How do you determine if the equation #f(x) = 2(1/2)^x# represents exponential growth or decay?
• How do you determine if the equation #f(x) = 5(1/6)^x# represents exponential growth or decay?
• How do you determine if the equation #f(x) = .08^x# represents exponential growth or decay?
• How do you determine if the equation #y = 30(2^x)# represents exponential growth or decay?
• The population of starlings in Lower Fremont was 20,000 in 1962. In 2004 the population is 160,000. How do you calculate the percentage rate of starling population growth in Lower Fremont since 1962?
• The population of rabbits in East Fremont is 250 in September of 2004, and growing at a rate of 3.5% each month. If the rate of population growth remains constant, determine the month and year in which the rabbit population will double?
• The population of rabbits in East Fremont is 250 in September of 2004, and growing at a rate of 3.5% each month. If the rate of population growth remains constant, determine the month and year in which the rabbit population will reach 128,000?
• In 2004, the population of butterflies in West Fremont is 60,000, and growing at a rate of 1.4% each year. If the rate of population growth remains constant, determine the year in which the population will double?
• In 2004, the population of butterflies in West Fremont is 60,000, and growing at a rate of 1.4% each year. If the rate of population growth remains constant, determine the year in which the population will reach 240,000?
• A culture of bacteria is growing at a constant rate of 5% an hour. If the culture started with 3000 bacteria at 6:00 a.m. on Tuesday, calculate the approximate time that the bacteria population will reach 72,000?
• The value of a $1,200 computer decreases 27% annually. What will be the value of the computer after 3 years?
• How do you determine if the equation #y=39(0.098)^t# represents exponential growth or decay?
• A radioactive substance has a half life of 120 years. Presently, there are 60 grams of the substance. When were 600 grams present?
• The cost of a first-class postage stamp in 1962 was 4. In 1999 it is 33. The cost was (and is) growing exponentially. What was the growth rate?
• A radioactive substance decays at a rate of 30% per year. Presently there are 400 grams of the substance. How do you find the amount left after 8 years?
• A radioactive substance decays at a rate of 30% per year. Presently there are 400 grams of the substance. How long before there are 10 grams?
• The count in a bacteria culture was 700 after 20 minutes and 1000 after 40 minutes. What was the initial size of the culture?
• A certain strain of bacteria grows from 40 to 326 in 120 minutes. How do you find k for the growth formula y=ae^(kt) where t, is the minutes?
• Carl plans to invest $500 at 8.25% interest, compounded continuously. How long will it take for his money to triple?
• How do you determine if the equation # y = -5(1/3) ^ -x# represents exponential growth or decay?
• In 1999 the worlds population reached 6 billion an was increasing at a rate of 1.3% per year. Assume that this growth rate remains constant, how do you write a formula for the world population (in billions) as a function of the number of years since 1999?
• How do you determine if the equation #y=90(5)^x# represents exponential growth or decay?
• How do you determine if the equation #y= 5(0.9)^x# represents exponential growth or decay?
• How do you determine if the equation #y= 90 (0.5)^x# represents exponential growth or decay?
• How do you determine if the equation #y= 0.5 (0.9)^x# represents exponential growth or decay?
• A population P is initially 3100 and four hours later reaches the given number. How do you find the four-hour growth or decay factor?
• A research assistant made 160 mg of radioactive sodium (Na^24) and found that there was only 20 mg left 45 h later, what is the half-life of Na^24?
• A research assistant made 160 mg of radioactive sodium (Na^24) and found that there was only 20 mg left 45 h later, how much of the original 20 mg would be left in 12 h?
• The population of a city was estimated to be 125 000 in 1930 and 500 000 in 1998, if the population continues to grow at the same rate when will the population reach 1 million?
• A small country that had 20 million people in 1990 has experienced exponential growth in population of 4% per year since then. How do you write an equation that models this situation and use your equation to determine when the population will double?
• How do you determine if the equation #y=(1/2)^x# represents exponential growth or decay?
• How do you determine if the equation #y=(1/4)^x# represents exponential growth or decay?
• How do you determine if the equation #y=2^x# represents exponential growth or decay?
• A bacteria culture starts with 500 bacteria and grows at a rate proportional to its size. After 3 hours there are 9,000 bacteria. How do you find the number of bacteria after 5 hours?
• How do you find the exponential growth function, if under ideal conditions, a population of rabbits has an exponential growth rate of 11.5% per day and it has an initial population of 900 rabbits?
• Bacteria grow by cell division. If each cell divides into 2 in every 2 minutes how many cells will exist after 32 minutes? Assume there was only one cell at the beginning?
• Assume that the number of bacteria follows an exponential growth model: . The count in the bacteria culture was 100 after 10 minutes and 1200 after 30 minutes. What was the initial size of the culture?
• Suppose you want to end up with $5,000 in a bank account after 4 years earning a rate of 3.5% compounding monthly. How much would you have to initially invest?
• A culture started with 2,000 bacteria. After 2 hours, it grew to 2,400 bacteria. Predict how many bacteria will be present after 10 hours?
• The population of Nigeria was about 140 million in 2008 and the exponential growth rate was 2.4% per year. How do you write an exponential function describing the population of Nigeria?
• The volume of harvestable timber in a young forest grows exponentially with a yearly rate of increase equal to 3.5%. What percentage increase is expected in 10 years?
• The half-life of a certain radioactive material is 85 days. An initial amount of the material has a mass of 801 kg. How do you write an exponential function that models the decay of this material and how much radioactive material remains after 10 days?
• A sailboat that costs $5950 decreases in value by 11% per year. How much will the boat be worth after 6 years?
• The half-life of a certain radioactive material is 75 days. An initial amount of the material has a mass of 381 kg. How do you write an exponential function that models the decay of this material and how much radioactive material remains after 15 days?
• For an annual rate of change of –31%, how do you find the corresponding growth or decay factor?
• The half life of Chemical Q is 12 days. If you started your experiment with 42 grams, how much would be left in 73 days?
• An initial population of 505 quail increases at an annual rate of 23%. How do you write an exponential function to model the quail population?
• The population of the US was 203 million in the year 1970 and 249 million in the year 1990. If it is growing exponentially, what will it be in the year 2030?
• The population of a cit grows at a rate of 5% each year. The population in 1990 was 400,000. What would be the predicted current population? In what year would we predict the population to reach 1,000,000?
• How do you determine if the equation #y = 4^x# represents exponential growth or decay?
• How do you determine if the equation #y = 2.1(0.15)^x# represents exponential growth or decay?
• How do you determine if the equation #y = 5(6.8)^x# represents exponential growth or decay?
• How do you determine if the equation #y = 2(3)^x# represents exponential growth or decay?
• How do you determine if the equation #y = 0.5(4)^x# represents exponential growth or decay?
• How do you determine if the equation #y = 4^x# represents exponential growth or decay?
• How do you determine if the equation #y=400(1.03)^x# represents exponential growth or decay?
• How do you determine if the equation # y = 0.84(1.45)^x# represents exponential growth or decay?
• How do you determine if the equation #y = 1.23(1.02)^x# represents exponential growth or decay?
• How do you determine if the equation #y = 0.84(0.94)^x# represents exponential growth or decay?
• How do you determine if the equation #y = 1.7(1.06)^x# represents exponential growth or decay?
• The Diaz family bought a house for $225,000. If the value of the house increases at a rate of 5% per year, about how much will the house be worth in 8 years?
• Lance puts $2500 in a savings account that pays 3.75% interest compounded yearly. How much money to the nearest dollar will be in the account 5 years later if he makes no more deposits or withdrawals?
• An investment is losing value at a rate of 2% per year. Your original investment was $1250 three years ago. How much is it worth now to the nearest dollar?
• The population of Detroit, Michigan was 951,300 in 2000. Detroit has been experiencing a population decline of 1.4% per year since 2000. What is the projected population for 2005 in Detroit?
• The population of a city is 50,000. It is increasing at an annual rate of 3%. What is the population of this city in 4 years?
• A patient takes one pill containing 50 mg each day and of that 40% will be washed out. How much of the medication is in the patient's system after 3 days?
• Sharon purchased land in 1970 for $40,000. That land increased in value approximately 3% per year. In 2005 Sharon decided to sell the land. How much could she reasonably ask as a selling price?
• A wealthy couple decided to start spending their fortune and spent half of what they had every 3 years. If they started with $20 million, how much would be left after 10 years?
• At a certain time, bacteria in a culture has 115,200 bacteria. Four hours later, the culture has 1,440,000 bacteria. How many bacteria will there be in the culture in 15 hours?
• How do you determine if the equation #y = (0.1)^x# represents exponential growth or decay?
• How do you determine if the equation #y = 2(1/4)^x# represents exponential growth or decay?
• How do you determine if the equation #y = (3)^x# represents exponential growth or decay?
• How do you determine if the equation #y = 1/2 (5)^x# represents exponential growth or decay?
• How do you determine if the equation #y = (2)^x# represents exponential growth or decay?
• How do you determine if the equation #y = 3(0.4)^x# represents exponential growth or decay?
• How do you determine if the equation #y=0.5(1.2)^x # represents exponential growth or decay?
• How do you determine if the equation #y=10(1.1)^x # represents exponential growth or decay?
• How do you determine if the equation #y=100(0.9)^x# represents exponential growth or decay?
• How do you determine if the equation #y=3(5/2)^x# represents exponential growth or decay?
• How do you determine if the equation #y=0.4(1/3)^x# represents exponential growth or decay?
• How do you determine if the equation #y=(0.5)^x# represents exponential growth or decay?
• The number of bacteria in a culture grew from 275 to 1135 in three hours. How do you find the number of bacteria after 7 hours?
• A $10,500 investment has a 15% loss each year. How do you determine the value of the investment after each of the following years: 1,2,4, and 10?
• How do you determine if the equation #y=15(7)^x# represents exponential growth or decay?
• A cheetah was chasing and antelope which is 700m away. How many minutes will it take the cheetah, which is running at a constant speed of 110km/h to reach the antelope that is running at a constant speed of 75km/h?
• How do you determine how much time is required for an investment to double in value if interest is earned at the rate of 6.25% compounded annually?
• A new car depreciates at a rate of 15% per year. What is the expected value of a $25,000 car after 5 years (rounded to nearest whole dollar)? A) $20750 B) $11093 C) $9429 D) $6250
• Question #6df28
• Question #52535
• How do you write a rule for this exponential growth function if one point is at (-1,4) and the other is (0,3)?
• At the start of an experiment, there are 100 bacteria. If the bacteria follow an exponential growth pattern with rate k = 0.02, what will be the population after 5 hours? How long will it take for the population to double?
• Question #cdeb4
• Question #9c59d
• Question #20ec7
• Why is it impossible for exponential growth to continue forever?
• Can anyone explain the logic behind why we used exponantial functions to model bacteria growth?
• Why is the area of a circle given by the formula #pir^2# ?
• Question #4099b
• Question #9415f
• How do you find the formula of the linear function described by the table #((t, 6.2, 6.4, 6.6, 6.8), (f(t), 606.4, 618.8, 631.2, 643.6))# ?
• Question #c3be8
• Population increases R% every year and after n years the population becomes P. What was/were the population before n years?
• Last year the income for a business was 2,000,000. This year the income grew 15%. What was the income for the business this year?
• Which function represents exponential growth? A. f(x)=(7/9)^-x B. f(x)=(5/6)^x C. f(x)=(8/3)^-x D.f(x)=-.56^x
• Question #b1d2e
• How do you simplify #(\frac { 2x ^ { 4} y ^ { 0} z ^ { - 1} } { x ^ { - 1} y ^ { 2} \cdot x y ^ { - 4} z ^ { 4} } ) ^ { - 4}#?
• Question #613e0
• Question #0074b
• How do you multiply and simplify #\frac { x ^ { 2} + 2x y + y ^ { 2} } { 2x ^ { 2} + 2x y } \cdot \frac { x - y } { x ^ { 2} - y ^ { 2} }#?
• Question #7f620
• Question #76a4c
• Question #19f52
• Question about exponential growth?