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Quizzes > High School Quizzes > Mathematics

Exponential Growth & Decay Practice Quiz

Master word problems on growth and decay concepts

Difficulty: Moderate
Grade: Grade 9
Study OutcomesCheat Sheet
Paper art themed algebra practice quiz for high school students on exponential growth and decay.

What is the formula for exponential growth in discrete compounding?
P = P0(1 + r)^t
P = P0 + r*t
P = P0*r^t
P = P0e^(rt)
The formula for exponential growth in discrete compounding is P = P0(1 + r)^t, where P0 is the initial amount, r is the growth rate, and t is the number of time periods. This formula shows the effect of compounding the growth rate over each period.
A bacteria colony doubles every hour. If the initial count is 100 bacteria, what is the population after 3 hours?
200
400
800
1000
The colony doubles every hour, so after 3 hours the population is calculated as 100 × 2^3, which equals 800. Each hour, the population multiplies by 2, resulting in exponential growth.
In the expression 5(0.8)^t, what does the exponent t represent?
The initial quantity
The number of time periods
The decay factor
The growth multiplier
In an exponential function, the exponent t represents the number of time periods over which the process occurs. It indicates how many times the base factor (in this case, 0.8) is applied.
Which statement best describes exponential decay?
The quantity decreases by a fixed amount each period
The quantity decreases by a constant percentage each period
The quantity remains constant over time
The quantity increases by a fixed amount each period
Exponential decay involves a constant percentage decrease per period rather than a fixed numerical amount. This means the quantity decreases proportionally over time.
If $500 depreciates at 10% per year, what is its value after 2 years?
$405
$450
$400
$350
Depreciation at 10% per year means the value is multiplied by 0.9 each year. After 2 years, the value is $500 × (0.9)^2, which equals $405.
A population grows from 200 to 400 in 3 years following the model A = P0(1 + r)^t. What is the approximate annual growth rate r?
26%
33%
20%
50%
Dividing the equation 400 = 200(1 + r)^3 gives (1 + r)^3 = 2. Taking the cube root of 2 results in approximately 1.26, so r is about 26%.
A car depreciates continuously according to V = V0e^(-kt). If a car worth $20,000 depreciates to $15,000 in 5 years, what is k approximately?
0.0575
0.0750
0.1000
0.0250
Dividing 15,000 by 20,000 gives 0.75, so e^(-5k) = 0.75. Taking the natural logarithm yields -5k = ln(0.75), and solving for k gives approximately 0.0575.
A town's population decreases by 5% annually. What is the decay factor used in its exponential model?
1.05
0.95
-0.05
0.05
A 5% decrease per year means that 95% of the population remains, giving a decay factor of 0.95. This factor is used to multiply the population each year.
If an investment doubles over 10 years according to A = A0(1 + r)^t, what is the approximate annual interest rate?
7.2%
10%
5%
3%
Since (1 + r)^10 = 2, taking the 10th root gives 1 + r ≈ 2^(1/10), or about 1.072. Therefore, the annual interest rate r is approximately 7.2%.
A radioactive substance has a half-life of 8 years. What fraction of the substance remains after 16 years?
1/4
1/2
1/8
3/4
After 16 years, which is two half-lives (16/8 = 2), the remaining fraction is (1/2)^2 = 1/4. This is the typical behavior of a half-life decay process.
If a population declines to 70% of its initial value in 3 years using P = P0e^(-kt), what is the approximate decay constant k?
0.119
0.300
0.070
0.050
Starting with 0.7 = e^(-3k), taking the natural logarithm gives -3k = ln(0.7). Solving for k yields approximately 0.119, which represents the decay rate per year.
Evaluate Q = 1000(1.05)^4. What is the approximate value of Q?
Approximately 1216
Approximately 1100
Approximately 1300
Approximately 1500
Calculating (1.05)^4 gives roughly 1.2155, so Q ≈ 1000 × 1.2155 ≈ 1216. This shows how compounded growth works over 4 periods.
An investment compounded annually at 8% will triple in value over time. Approximately how many years will this take?
About 14 years
About 10 years
About 20 years
About 7 years
Solving the equation (1.08)^t = 3 by taking logarithms yields t ≈ ln(3)/ln(1.08), which is approximately 14.3 years. This calculation demonstrates the time required for an investment to triple.
A substance decays by 2% per month. What is the monthly multiplication factor in its exponential decay model?
0.98
1.02
0.02
0.50
A 2% decrease means 98% of the quantity remains each month, so the multiplication factor is 0.98. This factor is consistently applied in each time period.
In the exponential model for bacteria growth, N = N0×2^t, what does t represent?
1 hour (the doubling time)
2 hours
0.5 hours
3 hours
In the model N = N0×2^t, the exponent t indicates the number of doubling periods, with each period being 1 hour. Therefore, t represents the number of hours elapsed.
A radioactive element decays from 80 grams to 10 grams in 12 years using N = N0e^(-kt). What is the approximate value of the decay constant k?
Approximately 0.1733
Approximately 0.2500
Approximately 0.1250
Approximately 0.5000
Dividing 10 by 80 gives 0.125, so e^(-12k) = 0.125. Taking logarithms leads to -12k = ln(0.125) = -ln(8), and solving yields k = ln(8)/12, which is approximately 0.1733 per year.
In a continuously compounded investment, $1000 grows to $1500 in 3 years. What is the annual rate r as a percentage?
Approximately 13.5%
Approximately 15%
Approximately 10%
Approximately 20%
Using the formula A = P0e^(rt) with 1500 = 1000e^(3r), solving for r gives r = ln(1.5)/3, which is approximately 0.1352 or 13.5%. This is the continuous rate needed for the investment to grow to $1500 in 3 years.
A cell culture triples in 5 days, following continuous growth. How many days will it take for the culture to increase tenfold?
Approximately 10.48 days
Approximately 15 days
Approximately 8 days
Approximately 12 days
From e^(5k) = 3, we find k = ln(3)/5. To achieve a tenfold increase, solve e^(kt) = 10, giving t = ln(10)/k = 5 ln(10)/ln(3), which is approximately 10.48 days. This method uses logarithms to relate growth factors and time.
A tree's height grows exponentially. If it is 2 meters tall now and is expected to be 8 meters in 20 years, what is its annual growth rate r?
Approximately 7.18%
Approximately 10%
Approximately 5%
Approximately 3%
Using the model H = H0(1 + r)^t with 8 = 2(1 + r)^20, we have (1 + r)^20 = 4. Taking the 20th root yields 1 + r = 4^(1/20), so r = 4^(1/20) - 1, which is approximately 7.18%.
Using the model P = P0(1 + r)^t, if a population increases by 20% over 4 years, what is the approximate annual growth rate?
Approximately 4.66%
Approximately 5%
Approximately 3.5%
Approximately 6%
Since (1 + r)^4 = 1.20, taking the fourth root gives 1 + r = (1.20)^(1/4) which is approximately 1.0466. Thus, the annual growth rate r is about 4.66%.
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Study Outcomes

  1. Analyze exponential growth and decay scenarios to identify key components of the problems.
  2. Apply algebraic methods to solve exponential growth and decay word problems.
  3. Interpret and formulate models that explain real-world exponential changes.
  4. Evaluate the effects of varying parameters within exponential functions.
  5. Synthesize learned concepts to predict future values in growth and decay contexts.

Exponential Growth/Decay Word Problems Cheat Sheet

  1. General Form of Exponential Functions - At its core, an exponential function looks like y = a × ekt. The "a" jump‑starts your value, while "k" decides if you're blasting off (positive) or winding down (negative). Just plug in time "t" and watch the magic unfold! Exponential Growth and Decay
  2. Identifying Growth vs. Decay - Growth is a party when the base is above 1, powering your values upward, while decay happens when it dips between 0 and 1, sneaking them downward. Spotting this difference is key to cracking exponential riddles and picking the right formula for the job! Exponential Growth and Decay
  3. Calculating the Growth/Decay Factor - When you hear "5% growth," transform that percent into its decimal sidekick (0.05) and add it to 1 for a factor of 1.05. For decay, subtract instead - and presto, you've got your shrinkage factor. This simple trick makes plugging numbers into formulas feel like child's play! Exponential Growth and Decay Practice
  4. Real‑World Problem Practice - Dive into scenarios like populations skyrocketing, radioactive atoms vanishing, or money multiplying with compound interest. Real examples turn abstract formulas into relatable stories and boost your confidence. Practice makes perfect, so grab your calculator and get messy! Exponential Growth and Decay Practice
  5. Understanding Half‑Life - In decay land, half‑life is the coolest stopwatch: it measures how long it takes for a substance to halve itself. Whether you're tackling nuclear physics or drug elimination in biology, half‑life gives you a natural timer. Mastering this concept is like having a secret weapon for science problems! Exponential Growth and Decay
  6. Using Natural Logarithms - Need to solve for "t" or "k"? Call in the natural logarithm! ln(x) undoes ex like a boss, letting you isolate time or rate with ease. Think of it as the reverse gear for your exponential engine. Exponential Growth and Decay
  7. Key Graph Features - Exponential graphs come with signature traits: horizontal asymptotes, flashy intercepts, and that swoopy curve. Recognizing each feature helps you sketch and interpret behaviors at a glance. Next time you see a graph, say "I know you!" and ace the question. Exponential Growth and Decay Practice
  8. Converting Between Forms - Switching between y = a × bt and y = a × ekt is like changing outfits for the same model - just update the rate using k = ln(b). This flexibility means you'll never be caught off guard on an exam! Exponential Growth and Decay Practice
  9. Impact of the Initial Value - Never underestimate "a" - it sets your starting line and shapes how steep your curve climbs or dives. A big "a" gives a bold launchpad, while a small "a" keeps things cozy near the axis. Understanding this lets you predict exactly where your graph begins! Exponential Growth and Decay
  10. Applications Across Fields - From compound interest fattening your bank account to bacteria booming in a petri dish, exponential functions show up everywhere. Spotting these patterns in finance, biology, or physics makes math feel more like a treasure hunt. Embrace exponentials and become a real‑world wizard! Exponential Growth and Decay
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