We need to find two things:
1. The function representing the store owner's profit after [tex]\( t \)[/tex] years considering an exponential growth.
2. The monthly percentage rate of change, rounded to the nearest hundredth of a percent.
### Step 1: Determine the Exponential Growth Function
Given:
- The average daily profit [tex]\( P_0 = \$ 470 \)[/tex].
- The annual growth rate [tex]\( r = 87\% = 0.87 \)[/tex].
The profit after [tex]\( t \)[/tex] years can be modeled by an exponential growth function:
[tex]\[ P(t) = P_0 \cdot (1 + r)^t \][/tex]
Plugging in the values we have:
[tex]\[ P(t) = 470 \cdot (1 + 0.87)^t \][/tex]
### Step 2: Convert Annual Growth Rate to Monthly
The annual growth rate is 87%, but we need to find the equivalent monthly growth rate.
To convert the annual growth rate to a monthly growth rate [tex]\( r_m \)[/tex], we use the formula:
[tex]\[ (1 + r) = (1 + r_m)^{12} \][/tex]
Solving for [tex]\( r_m \)[/tex]:
[tex]\[ 1 + 0.87 = (1 + r_m)^{12} \][/tex]
Taking the twelfth root of both sides:
[tex]\[ 1.87^{1/12} = 1 + r_m \][/tex]
Now, let's calculate [tex]\( r_m \)[/tex]:
[tex]\[ r_m = 1.87^{1/12} - 1 \][/tex]
Calculating this, we get:
[tex]\[ r_m \approx 0.054604 \][/tex]
So, the monthly growth rate [tex]\( r_m \)[/tex] is approximately 0.054604.
### Step 3: Monthly Growth Rate as a Percentage
To express the monthly growth rate as a percentage, we multiply by 100:
[tex]\[ r_m \times 100 \approx 0.054604 \times 100 \approx 5.46\% \][/tex]
### Step 4: Round the Coefficients in the Function
The coefficients in our function should be rounded to four decimal places.
Our final function is:
[tex]\[ P(t) = 470 \cdot (1.8700)^t \][/tex]
This represents the store owner's profit after [tex]\( t \)[/tex] years.
### Final Answer
- The exponential profit function is [tex]\( P(t) = 470 \cdot (1.8700)^t \)[/tex].
- The monthly rate of change is approximately [tex]\( 5.46\% \)[/tex].
So,
[tex]\[ f(t) = 470 (1.8700)^t \][/tex] where the monthly rate of change is [tex]\( 5.46\% \)[/tex].
