Exponentials Basics
State the initial value given the following equation:
y=4600(1.2)^x
4600
A town has a population of 12,000 and grows at 5% every year. What will the population be after 6 years, to the nearest whole number?
1200(1.05)6=16081 people
Write an exponential function in the form y=abx that goes through the points (0,18) and (3,144).
y=18(2)x
A population of rabbits is reintroduced to a state park. Initially, the park releases 45 rabbits. After one year, they discovered the population had doubled. How many rabbits would be expected after 5 years?
45(2)5=1440 rabbits
Tallulah invested $70,000 in an account paying an interest rate of 5.2% compounded quarterly. Assuming no deposits or withdrawals are made, how much money would be in the account after 13 years? Round to the nearest cent.
70000(1+0.052/4)^(4*13)=$137,021.65
Describe a possible scenario for the following equation:
f(t)=770(1.045)^t
An account starts at $770 and gains 4.5% per year.
Identify if the change is growth or decay and the percentage rate of increase or decrease.
y=10(1.031)^x
Growth by 3.1%
A new car is purchased for $23,400. The value of the car depreciates at 14% per year. What will the value of the car be after 13 years, to the nearest cent?
23400(0.86)13=$3293.79
Write an exponential function in the form y=abx that goes through the points (0,11) and (2,891).
y=11(9)x
A radioactive substance has a starting mass of 147 grams and a half-life of one year. What is the substance's mass after 7 years, rounded to the nearest hundredth?
147(0.5)7=1.15 grams
Jeriel invested $4200 in a certificate of deposit (CD) at his bank. It pays 5.5% compounded monthly. Assuming no other deposits or withdrawals are made, how much money would be in the account after 7 years? Round to the nearest cent.
4200(1+0.055/12)^(12*7)=$6166.95
Describe a possible scenario for the following equation:
f(x)=25900(0.93)^x
The car's value starts at $25,900 and depreciates by 7% each year.
Identify if the change is growth or decay and the percentage rate of increase or decrease.
y=340(0.904)^x
Decay by 9.6%
$3400 is placed in an account with an annual interest rate of 5.5%. How much will be in the account after 28 years, to the nearest cent?
3400(1.055)^28=$15224.67
Write an exponential function in the form y=abx that goes through the points (0,8200) and (4,9800).
y=8200(1.046)x
In a lab experiment, 9200 bacteria are placed in a petri dish. The conditions are such that the number of bacteria is able to double every 29 hours. How many bacteria would there be after 25 hours, to the nearest whole number?
9200(2)^(25/29)=16722
bacteria
Marjorie invested $44,000 in an account paying an interest rate of 5.2% compounded continuously. Assuming no other deposits or withdrawals are made, how much money would be in the account after 20 years? Round to the nearest cent.
44000e^(0.052*20)=$124,485.55
Describe a possible scenario for the following equation:
g(x)=4500(1+0.06/12)^(12*x)
An account starts at $4500 and earns interest at a rate of 6% compounded monthly.
Identify if the change is growth or decay and the percentage rate of increase or decrease.
y=3.5(1.974)^x
Growth by 97.4%
An element with mass 160 grams decays by 14.3% per minute. How much of the element is remaining after 10 minutes? Round to the nearest hundredth.
160(0.857)10=34.19 grams
The population of a now endangered animal on an island is decaying exponentially. In the year 2009, the population was 2600, and by 2012 the it was only 1030. Write an equation to represent the number of animals, A, over time, t, in years.
y=2600(0.734)x
A person invested $800 in an account growing at a rate allowing the money to double every 14 years. How much money would be in the account after 26 years, to the nearest cent?
800(2)^(26/14)=$2898.32
The metro are population of Dubai was 3,008,000 in 2023. Assuming the continuous growth rate stays at 1.42%, what is the approximate population in 2026? Round to the nearest whole number.
3,008,000e^(0.0142*3)=3,138,909
people
Describe a possible scenario for the following equation:
g(t)=350000e^(0.1t)
The population of a city starts at 350,000 people and is continuously growing at a rate of 10% per year.
Identify if the change is growth or decay and the percentage rate of increase or decrease.
y=900(0.2)^x
Decay by 80%
The temperature of a cup of hot chocolate straight out of the microwave has a temperature of 160oF. It begins to cool at a rate of 5.19% per minute. What temperature will the hot chocolate be after 5 minutes? Round to the nearest hundredth.
160(0.9481)5=122.57oF
Zachary purchased a new car in 2004 for $19,500. The value of the car has been depreciating exponentially at a constate percentage rate. If the value of the car was $13,300 in the year 2008, write an equation for the value of the car, V, in terms of t years.
V=19500(0.909)x
A new car is purchased for $50,000 and over time its value depreciates by one half every 3 years. What is the value of the car 23 years after it was purchased, to the nearest penny?
50000(1/2)^(23/3)=$246.08
A radioactive isotope starts with a mass of 268 grams and continuously decays at a rate of 14.9% per year. What is the mass after 6 years? Round to the nearest hundredth.
268e^(-0.149*6)=109.62
grams
Describe a possible scenario for the following equation:
f(t)=103e^(-0.05t)
The temperature of bath water starts at 103oF and continuously cools down at a rate of 5% per minute.