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Answer :
Let's solve the problem step-by-step to find the approximate value of [tex]\( P \)[/tex].
We have the function [tex]\( f(t) = P \cdot e^{r \cdot t} \)[/tex]. We're given that [tex]\( f(4) = 246.4 \)[/tex]. We also know [tex]\( r = 0.04 \)[/tex] and [tex]\( t = 4 \)[/tex].
Our goal is to find the approximate value of [tex]\( P \)[/tex].
### Step 1: Write the Equation for [tex]\( f(4) \)[/tex]
Given:
[tex]\[
f(4) = P \cdot e^{0.04 \cdot 4}
\][/tex]
### Step 2: Substitute the Known Value
We know [tex]\( f(4) = 246.4 \)[/tex], so:
[tex]\[
246.4 = P \cdot e^{0.04 \cdot 4}
\][/tex]
### Step 3: Simplify the Exponent
Calculate the exponent:
[tex]\( 0.04 \cdot 4 = 0.16 \)[/tex]
So, the equation becomes:
[tex]\[
246.4 = P \cdot e^{0.16}
\][/tex]
### Step 4: Solve for [tex]\( P \)[/tex]
To find [tex]\( P \)[/tex], rearrange the equation:
[tex]\[
P = \frac{246.4}{e^{0.16}}
\][/tex]
### Step 5: Calculate [tex]\( e^{0.16} \)[/tex]
Using a calculator:
[tex]\[
e^{0.16} \approx 1.1735
\][/tex]
### Step 6: Calculate [tex]\( P \)[/tex]
Divide 246.4 by 1.1735:
[tex]\[
P \approx \frac{246.4}{1.1735} \approx 209.97
\][/tex]
### Step 7: Choose the Closest Option
The closest value among the options is [tex]\( P \approx 210 \)[/tex].
So, the approximate value of [tex]\( P \)[/tex] is:
A. 210
We have the function [tex]\( f(t) = P \cdot e^{r \cdot t} \)[/tex]. We're given that [tex]\( f(4) = 246.4 \)[/tex]. We also know [tex]\( r = 0.04 \)[/tex] and [tex]\( t = 4 \)[/tex].
Our goal is to find the approximate value of [tex]\( P \)[/tex].
### Step 1: Write the Equation for [tex]\( f(4) \)[/tex]
Given:
[tex]\[
f(4) = P \cdot e^{0.04 \cdot 4}
\][/tex]
### Step 2: Substitute the Known Value
We know [tex]\( f(4) = 246.4 \)[/tex], so:
[tex]\[
246.4 = P \cdot e^{0.04 \cdot 4}
\][/tex]
### Step 3: Simplify the Exponent
Calculate the exponent:
[tex]\( 0.04 \cdot 4 = 0.16 \)[/tex]
So, the equation becomes:
[tex]\[
246.4 = P \cdot e^{0.16}
\][/tex]
### Step 4: Solve for [tex]\( P \)[/tex]
To find [tex]\( P \)[/tex], rearrange the equation:
[tex]\[
P = \frac{246.4}{e^{0.16}}
\][/tex]
### Step 5: Calculate [tex]\( e^{0.16} \)[/tex]
Using a calculator:
[tex]\[
e^{0.16} \approx 1.1735
\][/tex]
### Step 6: Calculate [tex]\( P \)[/tex]
Divide 246.4 by 1.1735:
[tex]\[
P \approx \frac{246.4}{1.1735} \approx 209.97
\][/tex]
### Step 7: Choose the Closest Option
The closest value among the options is [tex]\( P \approx 210 \)[/tex].
So, the approximate value of [tex]\( P \)[/tex] is:
A. 210
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