Question

the exponential model below represents the population of a animal x years from now. what is meaning of 0.94 in the model?
$f(x) = 8274(0.94)^x$
show your work here

• every year, the animal decreases in population by 94%
• the animal’s population will be zero after 6 years
• the animal will have doubled in population after 6 years
• every year, the animal increases in population by 94%
• every year, the animal increases in population by 6%
• every year, the animal decreases in population by 6%

Explanation:

Step1: Recall exponential decay model

The general form of an exponential decay model is \( f(x)=a(b)^x \), where \( a \) is the initial amount, \( b \) is the decay factor (\( 0 < b < 1 \)), and \( x \) is time. The decay rate \( r \) is related to \( b \) by \( b = 1 - r \).

Step2: Analyze the given function

For \( f(x)=8274(0.94)^x \), here \( b = 0.94 \). Calculate the decay rate: \( r=1 - 0.94 = 0.06 \), which is \( 6\% \). Since \( b = 0.94<1 \), it's a decay model, meaning the population decreases by \( 6\% \) each year (because \( 1 - 0.94 = 0.06 = 6\% \) decrease).

Step3: Evaluate the options

• Option 1: "Every year, the animal decreases in population by 94%" is wrong because the decrease rate is \( 6\% \), not \( 94\% \).
• Option 2: The function \( f(x)=8274(0.94)^x \) will never be zero (exponential decay approaches zero but never reaches it), so this is wrong.
• Option 3: The function is decreasing, so it can't double, wrong.
• Option 4: \( b = 0.94<1 \), so it's a decrease, not an increase, wrong.
• Option 5: As calculated, \( r = 6\% \) decrease per year, so this is correct.

Answer:

Every year, the animal decreases in population by 6% (the last option: "Every year, the animal decreases in population by 6%")