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Answer :
Certainly! Let's solve the question step-by-step:
We are given the function [tex]\( f(t) = P \cdot e^{r \cdot t} \)[/tex], where we're asked to find the approximate value of [tex]\( P \)[/tex]. We know that:
- [tex]\( f(4) = 246.4 \)[/tex]
- [tex]\( r = 0.04 \)[/tex]
We need to determine [tex]\( P \)[/tex] for [tex]\( t = 4 \)[/tex].
### Step 1: Write the equation with known values
Plug the known values into the equation:
[tex]\[ f(4) = P \cdot e^{0.04 \cdot 4} \][/tex]
Since [tex]\( f(4) = 246.4 \)[/tex], the equation becomes:
[tex]\[ 246.4 = P \cdot e^{0.16} \][/tex]
### Step 2: Solve for [tex]\( P \)[/tex]
To find [tex]\( P \)[/tex], we need to isolate it on one side of the equation:
[tex]\[ P = \frac{246.4}{e^{0.16}} \][/tex]
### Step 3: Calculate the exponential part
The value of [tex]\( e^{0.16} \)[/tex] is approximately 1.1735.
### Step 4: Calculate [tex]\( P \)[/tex]
Now, divide 246.4 by 1.1735 to get [tex]\( P \)[/tex]:
[tex]\[ P \approx \frac{246.4}{1.1735} \approx 209.97 \][/tex]
Therefore, the approximate value of [tex]\( P \)[/tex] is 210.
### Conclusion
The closest answer choice to 209.97 is option D: 210. Hence, the approximate value of [tex]\( P \)[/tex] is 210.
We are given the function [tex]\( f(t) = P \cdot e^{r \cdot t} \)[/tex], where we're asked to find the approximate value of [tex]\( P \)[/tex]. We know that:
- [tex]\( f(4) = 246.4 \)[/tex]
- [tex]\( r = 0.04 \)[/tex]
We need to determine [tex]\( P \)[/tex] for [tex]\( t = 4 \)[/tex].
### Step 1: Write the equation with known values
Plug the known values into the equation:
[tex]\[ f(4) = P \cdot e^{0.04 \cdot 4} \][/tex]
Since [tex]\( f(4) = 246.4 \)[/tex], the equation becomes:
[tex]\[ 246.4 = P \cdot e^{0.16} \][/tex]
### Step 2: Solve for [tex]\( P \)[/tex]
To find [tex]\( P \)[/tex], we need to isolate it on one side of the equation:
[tex]\[ P = \frac{246.4}{e^{0.16}} \][/tex]
### Step 3: Calculate the exponential part
The value of [tex]\( e^{0.16} \)[/tex] is approximately 1.1735.
### Step 4: Calculate [tex]\( P \)[/tex]
Now, divide 246.4 by 1.1735 to get [tex]\( P \)[/tex]:
[tex]\[ P \approx \frac{246.4}{1.1735} \approx 209.97 \][/tex]
Therefore, the approximate value of [tex]\( P \)[/tex] is 210.
### Conclusion
The closest answer choice to 209.97 is option D: 210. Hence, the approximate value of [tex]\( P \)[/tex] is 210.
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