SAT exponential growth and decay models practice

Population of a city doubles every 10 years. What is the growth model?

Exponential growth model. Formula: $\displaystyle P(t) = P_0 e^{rt}$, where $\displaystyle P_0$ is the initial population.

True or False: Investment growth is linear.

False. Investment growth is exponential due to compound interest.

If you invest $1,000 at 5% annual interest, how much in 10 years?

$\displaystyle 1,628.89 using the formula$A = P(1 + r)^t$.

Fill in the blank: A population grows exponentially if it grows by ____ each year.

A constant percentage.

How does a tree population grow in a forest over time?

Exponentially, as each tree produces seeds that grow into new trees.

Comparison: Exponential vs. Linear growth.

Exponential grows faster over time, while linear increases by a constant amount.

In a bacteria culture, count doubles every hour. After 3 hours, start with 100. How many?

800 bacteria. Growth: $\displaystyle 100 imes 2^3 = 800$.

True or False: Compounding results in faster growth than simple interest.

True. Compounding allows interest to earn interest.

What is the formula for population growth?

$\displaystyle P(t) = P_0 e^{rt}$ where $\displaystyle P_0$ is initial population, $\displaystyle r$ is growth rate.

If a car's value drops by 20% yearly, what type of decay is this?

This is an exponential decay model.

How much will an investment of $5000 grow in 5 years at 8%?

$\displaystyle 7,347.78 using$A = P(1 + r)^t$.

Fill in the blank: Exponential growth leads to ____ increases over time.

Dramatic or rapid.

After 4 years, a $\displaystyle 2000 investment grows to$2500. What is the approximate annual growth rate?

About 5.68%. Use $\displaystyle A = P(1 + r)^t$.

What is a real-world example of exponential growth?

The spread of a viral infection, where each infected person spreads it to multiple others.

True or False: The interest earned on savings accounts is linear.

False. It is usually compounded, leading to exponential growth.

If a population grows from 100 to 400 in 3 years, what can you infer?

It exhibits exponential growth. Doubling occurred multiple times.

True or False: A car's value decreases exponentially over time.

True. A car typically depreciates in value rapidly at first, then more slowly.

Fill in the blank: A radioactive substance decays by ___ each hour.

a fixed percentage of its remaining quantity.

Scenario: A couch worth $1,000 depreciates by 20% yearly. What is its value after 1 year?

Value after 1 year: $\displaystyle 800. Calculation:$1,000 - ($\displaystyle 1,000 imes 0.20) =$800.

Cause → Effect: Why does a smartphone lose value over time?

New models are released, causing older ones to depreciate.

What happens to the value of a car after 5 years if it loses 15% each year?

It decreases significantly: approximately $\displaystyle 422.50 if starting at$20,000.

Comparison: Depreciation vs. Radioactive decay

Both decrease over time, but depreciation is based on market value, while radioactive decay is based on half-lives.

A laptop originally priced at $800 loses 25% value annually. Calculate its value after 2 years.

Value after 2 years: $\displaystyle 450. Calculation:$800 - ($\displaystyle 800 imes 0.25) =$600 and then $\displaystyle 600 - ($600 imes 0.25) = $450.

True or False: A bicycle's value doubles each year.

False. A bicycle typically depreciates in value, not appreciates.

What is the annual decay rate if a $\displaystyle 2,000 item is worth$1,500 after 1 year?

Decay rate is 25%. Calculation: $\displaystyle 2,000 -$1,500 = $\displaystyle 500;$500/$2,000 = 0.25.

If a car is worth $30,000 and depreciates by 10% every year, what's its value after 3 years?

Approximately $\displaystyle 24,300. Calculation:$30,000 × (0.90)^3.

Fill in the blank: The half-life of a substance is the time it takes for ___ to decay.

half of the initial amount.

Scenario: A furniture item worth $500 depreciates 30% in the first year. What's its value?

$\displaystyle 350. Calculation:$500 - ($\displaystyle 500 imes 0.30) =$350.