Question

e. i can explain what a growth factor means in a situation.
a. jimmy takes 400 mg of advil. it decreases at a rate of 4% out of his body.
b. tara bought a car for $50,000 dollars. it decreases in value by 1.3% each year.
c. genovia’s population growth can be shown by the exponential function below.
$f(x) = 30000(1.20)^x$
d. the bacteria population in a petri dish can be calculated using the exponential function below.
$f(x) = 100(0.30)^x$

Explanation:

Step1: Analyze part a (Advil problem)

The general form for exponential decay is $y = a(1 - r)^t$, where $a$ is the initial amount, $r$ is the rate of decrease, and $(1 - r)$ is the decay factor (similar to growth factor but for decay). Here, initial amount $a = 400$ mg. The rate of decrease $r = 4\%=0.04$. So the decay factor (which is like a "growth factor" for decay, representing the remaining proportion) is $1 - 0.04 = 0.96$. This means each hour (or time unit), 96% of the Advil remains in his body. The initial value $a = 400$ mg means Jimmy initially took 400 mg of Advil.

Step2: Analyze part b (Car value problem)

Using the exponential decay formula $y = a(1 - r)^t$, initial amount $a = 50000$ dollars. Rate of decrease $r = 1.3\% = 0.013$. Decay factor is $1 - 0.013 = 0.987$, meaning each year the car's value is 98.7% of its previous year's value. Initial value $a = 50000$ dollars is the price Tara bought the car for.

Step3: Analyze part c (Population growth)

The general exponential growth formula is $y = a(b)^t$, where $a$ is initial value, $b$ is growth factor ($b>1$ for growth). For $f(x)=30000(1.20)^x$, initial value $a = 30000$, so Genovia's initial population is 30000. Growth factor $b = 1.20$, meaning the population increases by 20% each time period (since $1.20 = 1 + 0.20$), so the population is 120% of the previous period's population.

Step4: Analyze part d (Bacteria population)

The formula is $f(x)=100(0.30)^x$. Using exponential decay (since $0.30<1$) formula $y = a(b)^t$, initial value $a = 100$, so initially there are 100 bacteria. The factor $b = 0.30$, which is a decay factor (since it's less than 1), meaning each time period, the bacteria population is 30% of the previous period's population (so it's decaying, 70% decrease each period).

Answer:

Part a:

• Growth/Decay Factor: $0.96$ (decay factor), meaning 96% of the Advil remains in his body per time unit.
• Initial Value: $400$ mg, meaning Jimmy initially took 400 mg of Advil.

Part b:

• Growth/Decay Factor: $0.987$ (decay factor), meaning the car’s value is 98.7% of its previous year’s value.
• Initial Value: $\$50,000$, meaning Tara bought the car for $\$50,000$ initially.

Part c:

• Growth Factor: $1.20$, meaning the population increases by 20% per time period (population is 120% of the previous period’s population).
• Initial Value: $30,000$, meaning Genovia’s initial population was 30,000.

Part d:

• Growth/Decay Factor: $0.30$ (decay factor), meaning the bacteria population is 30% of the previous period’s population (70% decrease per period).
• Initial Value: $100$, meaning there were initially 100 bacteria.