Question

Determine whether the function below is exponential growth or exponential decay, and find the percentage rate of change.
P(t)= 8.5(0.92)^t


A
Exponential decay; 8.5%

B
Exponential decay; 8%

C
Exponential growth; 8%

D
Exponential growth; 8.5%

Answer 1

To determine whether the function \( P(t) = 8.5(0.92)^t \) represents exponential growth or decay, we need to look at the base of the exponent, which is \( 0.92 \).

- If the base is greater than 1, the function represents exponential growth.
- If the base is between 0 and 1, the function represents exponential decay.

Since \( 0.92 < 1 \), the function represents **exponential decay**.

Next, we can find the percentage rate of change. The general form for an exponential decay function is \( P(t) = P_0(1 - r)^t \), where \( r \) is the decay rate.

In our function, we can see that:

\[
P(t) = 8.5(0.92)^t
\]

Here, \( 0.92 = 1 - r \), which implies:

\[
r = 1 - 0.92 = 0.08
\]

To express this as a percentage:

\[
r = 0.08 \times 100\% = 8\%
\]

Putting this all together:

- The function is **exponential decay**.
- The percentage rate of change is **8%**.

Thus, the correct answer is:

**B. Exponential decay; 8%**.