Question

alg cc unit 4 lt25
determine whether the function below is exponential growth or exponential decay, and find the percentage rate of change.
p(t) = 3.5(0.09)^t
a exponential growth; 9%
b exponential growth; 91%
c exponential decay; 9%
d exponential decay; 91%

Explanation:

Step1: Recall exponential function form

The general form of an exponential function is \( P(t) = a(b)^t \), where \( a \) is the initial amount and \( b \) determines growth or decay. If \( b > 1 \), it's growth; if \( 0 < b < 1 \), it's decay. The percentage rate of change is \( (b - 1) \times 100\% \) (for growth, positive; for decay, negative, but we report the magnitude of the decay rate as a percentage decrease).

Step2: Analyze \( P(t) = 3.5(0.09)^t \)

Here, \( b = 0.09 \). Since \( 0.09 < 1 \), this is exponential decay. Now, calculate the percentage rate of change: \( (0.09 - 1) \times 100\% = (-0.91) \times 100\% = -91\% \). The negative sign indicates decay, and the percentage rate of decay is \( 91\% \) (since the quantity is decreasing by \( 91\% \) each time period, as \( 100\% - 9\% = 91\% \) decrease? Wait, no—wait, \( b = 0.09 \) means the remaining percentage is \( 9\% \), so the decay rate is \( 100\% - 9\% = 91\% \) decay. Wait, let's re - check: If \( P(t)=a(b)^t \), and \( b = 1 - r \) (for decay, where \( r \) is the decay rate as a decimal), then \( 0.09=1 - r\), so \( r = 1 - 0.09 = 0.91 \), so the percentage decay rate is \( 0.91\times100\% = 91\% \).

Answer:

D. Exponential decay; 91%