### Step-by-Step Solution
Let's address and solve the given problems one by one.
#### Problem 1: Interior Designer's Fee
The problem states that an interior designer charges a one-time initial fee of \[tex]$100 and then \$[/tex]15 for each hour of service.
The fee function is given by:
[tex]\[ f(x) = 15x + 100 \][/tex]
where [tex]\( x \)[/tex] is the number of hours of service.
The question does not specify any specific calculation for this function, but we have analyzed it numerically for [tex]\( x = 5 \)[/tex] hours.
1. The overall fee can be calculated as:
[tex]\[ f(5) = 15(5) + 100 = 75 + 100 = 175 \][/tex]
Thus, for [tex]\( x = 5 \)[/tex] hours, the total fee is \[tex]$175.
#### Problem 2: Bacteria Population Change
Two exponential functions model the population change of bacteria over time \( t \).
1. Exponential Decay:
\[
A. \quad P(t) = 500(0.8)^t
\]
This represents a model where the bacteria population is decreasing over time.
2. Exponential Growth:
\[
B. \quad P(t) = 250(1.25)^t
\]
This represents a model where the bacteria population is increasing over time.
We need to examine these functions at \( t = 5 \) hours.
- For the exponential decay model (\( P(t) = 500(0.8)^t \)):
\[
P(5) = 500(0.8)^5 = 163.84
\]
So, the bacteria population after 5 hours is approximately 163.84.
- For the exponential growth model (\( P(t) = 250(1.25)^t \)):
\[
P(5) = 250(1.25)^5 = 762.939453125
\]
So, the bacteria population after 5 hours is approximately 762.94.
### Summary of Results
- Interior Designer Fee:
For \( x = 5 \) hours, the total fee is \$[/tex]175.
- Bacteria Population:
- Exponential Decay: After 5 hours, the population is [tex]\( 163.84 \)[/tex].
- Exponential Growth: After 5 hours, the population is [tex]\( 762.94 \)[/tex].
These values summarize the results of the given models for the specified scenarios.
