Let's solve the problem step-by-step.
1. Finding the Initial Population Size:
The initial population size is the number of bacteria cells at time [tex]\( t = 0 \)[/tex]. The function given is:
[tex]\[ P(t) = 1500(0.92)^t \][/tex]
To find the initial population size, we substitute [tex]\( t = 0 \)[/tex] into the function:
[tex]\[ P(0) = 1500(0.92)^0 \][/tex]
We know that any number raised to the power of 0 is 1:
[tex]\[ P(0) = 1500 \cdot 1 \][/tex]
[tex]\[ P(0) = 1500 \][/tex]
So, the initial population size is [tex]\( \boxed{1500} \)[/tex].
2. Determining Growth or Decay:
Next, we need to determine if the function represents growth or decay. The behavior of the exponential function depends on the base of the exponent.
The base of the exponential function given is [tex]\( 0.92 \)[/tex]. Since [tex]\( 0.92 < 1 \)[/tex], the population is decreasing over time. Therefore, this function represents decay. We can conclude:
[tex]\[ \text{decay} \][/tex]
3. Calculating the Percent Change per Hour:
To find the percent change per hour, we analyze the base of the exponential function, which is [tex]\( 0.92 \)[/tex].
The percent change can be computed as follows:
[tex]\[ \text{Percent Change} = (0.92 - 1) \times 100\% \][/tex]
[tex]\[ \text{Percent Change} = -0.08 \times 100\% \][/tex]
[tex]\[ \text{Percent Change} = -8\% \][/tex]
Negative percent change indicates a decrease. Therefore, the population size decreases by [tex]\( \boxed{-8\%} \)[/tex] each hour.
