Writing Exponential Equations
In a game show, players start with 25 points and play multiple rounds to score additional points. Each round has 3 times as many points available as the previous round. Write an equation that shows the number of points available, P, in round r of the game show
P(r)=25(3)r
These equations represent the height of different magical trees over time, t, in years. Which equation represents the tree with the greatest growth factor?
A. H=3(1.25)t
B. H=2(1.5)t
C. H=4(1.75)t
D. H=5(1.4)tC - Growth factor: 1.75
A phone battery starts at 100% charge. Each hour, it loses 25% of its remaining charge.
Write an equation to model the battery level B after t hours.
B=100(0.75)t
An online game tracks its downloads with the equation d(t)=10,000(1.5)t where t is the number of days since it reached 10,000 downloads.
Find d(−2) and explain what it represents in this situation.
d(-2)=4,444
2 days before the game reached 10,000 downloads, it had about 4,444 downloads
The milligrams of caffeine remaining in a person's body is modeled by the equation C=200(1/2)t where t is the number of hours since the person drank coffee.
What does the 200 mean? What does 1/2 mean in this context?
200 --> started with 200 mg of caffeine
1/2 --> every hour, the caffeine amount is cut in half
A scientist observes a bacterial culture that doubles in number every hour. At the start, there are 40 bacteria. Write an equation that shows the number of bacteria, C, at hour h?
C(h)=40(2)h
These equations model the population of different animals in protected habitats over time, t, in years. Which one shows the population with the highest growth rate?
A. P=100(2)t
B. P=80(2.25)t
C. P=120(1.9)t
D. P=90(2.1)t
B - Growth factor: 2.25
A block of ice cream starts at 1,200 grams. Each minute, it loses 30% of its previous weight.
Write an equation for the remaining weight W of the ice cream after t minutes.
W=1200(0.7)t
A company models sales of a new gadget with the equation
s(t)=5,000(1.2)t where t is the number of days since the product launched.
Find s(−2) and explain what it means in the context of the product launch.
s(-2)=3,472
2 days before the product launched, there were approximately 3,472 units sold
The mass of a radioactive substance is modeled by the equation R=1,000(4/5)t where t is time in days.
What does the 1,000 represent? What does the 4/5 represent?
1,000 --> This is the initial mass of the radioactive substance
4/5 --> Each day, 80% of the substance remains, meaning it decreases by 20% daily.
A viral video gets 4 times as many views each day as it did the day before. On the day it was posted, it has 10 views. Write an equation to show the number of views, V, d days after it was posted.
V(d)=10(4)d
Each equation shows how an investment grows over time, t, in years. Which investment has the largest growth factor?
A. V=500(1.1)t
B. V=700(1.3)t
C. V=600(1.5)t
D. V=550(1.4)t
C - Growth factor: 1.5
A dose of medicine begins at 500 mg in the bloodstream. Every hour, the amount decreases to half of the previous amount.
Write an equation for the amount of medicine M left after t hours.
M=500(0.5)t
A social media influencer's followers on a new app are modeled by the equation f(t)=800(3)t where t is the number of days since they posted a viral video.
Find f(−2) and explain what it represents in this context.
f(-2)=89
2 days before the viral video was posted, the influencer had around 89 followers.
A water tank starts leaking, and the amount of water left in the tank is modeled by W=600(2/3)t where t is the number of hours since the leak started.
What does the 600 represent? What does the 2/3 represent?
600 --> There were 600 gallons of water in the tank at the moment the leak started.
2/3 --> Every hour, only 2/3 (≈66.7%) of the water remains, so the tank loses 1/3 of its water every hour.
An investment doubles in value every year. Initially, it's worth $1000. Write an equation that shows the investment value, V, after t years?
V(t)=1000(2)t
The following equations model how insect populations grow in different environments, where t is in weeks. Which insect population grows the fastest?
A. P=20(2.5)t
B. P=25(2.5)t
C. P=22(2.75)t
D. P=18(2.4)t
C - Growth factor: 1.5
A radioactive material has an initial mass of 800 grams. Every year, its mass decreases by 40%.
Write an equation to model the remaining mass R after t years.
R=800(0.60)t
The amount of algae in a tank is modeled by
a(t)=100(1.4)t where t is the number of days since a cleaning.
Find a(−2) and explain what it means in this context.
a(-2)=51
2 days before the tank was cleaned, the algae level would have been around 51 units
The amount of medicine in a child's body is given by M=120(3/4)t where t is the number of hours since the medicine was taken.
What does the 120 mean? What does 3/4 mean in this situation?
120 --> The child had 120 milligrams of medicine in their body when it was first taken.
3/4 --> Every hour, 75% of the medicine remains, meaning the medicine decreases by 25% each hour.
A magical tree quadrupoles in height every year. When it was planted, it was 2 feet tall. Write an equation that shows the height of the tree, H, after y years.
H(y)=2(4)y
Each function shows how contamination spreads in different lakes over time, t, in days. Which lake’s contamination spreads the most quickly?
A. C=10(1.6)t
B. C=12(1.9)t
C. C=8(2)t
D. C=11(1.85)t
C - Growth factor: 2
A car is worth $25,000 when new. Each year, its value decreases by 15%.
Write an equation to model the value of the car V after t years.
B=25000(0.85)t
A website tracks email subscribers using the equation e(t)=2,000(1.6)t where t is the number of days since a marketing campaign began.
Find e(−2) and explain what it represents in this situation.
e(-2)=781
2 days before the marketing campaign started, there were about 781 email subscribers.
The amount of fizz left in an open soda bottle is modeled by F=150(5/6)t where t is the number of minutes since the bottle was opened.
What does the 150 mean? What does 5/6 represent in this situation?
150 --> The soda started with 150 units of fizz (could be bubbles or carbonation level) when first opened.
5/6 --> Every minute, 5/6 (about 83.3%) of the fizz remains, so it loses about 16.7% of the fizz per minute.