Answer :
To find the approximate value of [tex]\( P \)[/tex] for the function [tex]\( f(t) = P \cdot e^{rt} \)[/tex], we need to work with the information given: [tex]\( f(3) = 191.5 \)[/tex] when [tex]\( r = 0.03 \)[/tex].
Here's a step-by-step solution:
1. Identify the Expression:
The function provided is [tex]\( f(t) = P \cdot e^{rt} \)[/tex].
2. Insert the Known Values:
We know [tex]\( f(3) = 191.5 \)[/tex], [tex]\( r = 0.03 \)[/tex], and [tex]\( t = 3 \)[/tex]. Substitute these into the function:
[tex]\[
191.5 = P \cdot e^{0.03 \cdot 3}
\][/tex]
3. Calculate the Exponential Part:
First, calculate [tex]\( e^{0.03 \cdot 3} = e^{0.09} \)[/tex]. The calculated value is approximately 1.0942.
4. Solve for [tex]\( P \)[/tex]:
Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{191.5}{e^{0.09}}
\][/tex]
Substituting the exponential value gives:
[tex]\[
P = \frac{191.5}{1.0942}
\][/tex]
This simplifies to approximately 175.
5. Determine the Answer:
The calculated value of [tex]\( P \)[/tex] is approximately 175, which matches option C.
Therefore, the approximate value of [tex]\( P \)[/tex] is:
C. 175
Here's a step-by-step solution:
1. Identify the Expression:
The function provided is [tex]\( f(t) = P \cdot e^{rt} \)[/tex].
2. Insert the Known Values:
We know [tex]\( f(3) = 191.5 \)[/tex], [tex]\( r = 0.03 \)[/tex], and [tex]\( t = 3 \)[/tex]. Substitute these into the function:
[tex]\[
191.5 = P \cdot e^{0.03 \cdot 3}
\][/tex]
3. Calculate the Exponential Part:
First, calculate [tex]\( e^{0.03 \cdot 3} = e^{0.09} \)[/tex]. The calculated value is approximately 1.0942.
4. Solve for [tex]\( P \)[/tex]:
Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{191.5}{e^{0.09}}
\][/tex]
Substituting the exponential value gives:
[tex]\[
P = \frac{191.5}{1.0942}
\][/tex]
This simplifies to approximately 175.
5. Determine the Answer:
The calculated value of [tex]\( P \)[/tex] is approximately 175, which matches option C.
Therefore, the approximate value of [tex]\( P \)[/tex] is:
C. 175
