Math Snap

STEP 1

1. The function $P(t) = 2500(0.82)^t$ represents the population size of bacteria after $t$ hours.
2. We need to determine the initial population size, whether the function represents growth or decay, and the percent change per hour.

STEP 2

1. Determine the initial population size.
2. Determine if the function represents growth or decay.
3. Calculate the percent change per hour.

STEP 3

The initial population size is the value of $P(t)$ when $t = 0$.
Substitute $t = 0$ into the function:
$$P(0) = 2500(0.82)^0$$ Since any number to the power of zero is 1:
$$P(0) = 2500 \times 1 = 2500$$ The initial population size is $2500$.

STEP 4

To determine if the function represents growth or decay, examine the base of the exponential function, which is $0.82$.
Since $0.82 < 1$, the function represents decay.

To find the percent change per hour, calculate the difference between the base and 1, and then convert it to a percentage.
The base is $0.82$, so the change is:
$$1 - 0.82 = 0.18$$ Convert $0.18$ to a percentage:
$$0.18 \times 100\% = 18\%$$ The population size decreases by $18\%$ each hour.
The initial population size is $2500$, the function represents decay, and the population size decreases by $18\%$ each hour.