Answer :
To solve this problem, we need to find the value of [tex]\( \rho \)[/tex] when given the function [tex]\( f(t) = \rho e^{rt} \)[/tex], where we know that [tex]\( f(4) = 246.4 \)[/tex] and [tex]\( r = 0.04 \)[/tex].
Here's a step-by-step breakdown of how to solve it:
1. Understand the Function: The function provided is [tex]\( f(t) = \rho e^{rt} \)[/tex]. This is an exponential function where [tex]\( \rho \)[/tex] is the initial value or the coefficient, [tex]\( r \)[/tex] is the growth rate, and [tex]\( t \)[/tex] is the time.
2. Substitute the Known Values: We have:
- [tex]\( f(4) = 246.4 \)[/tex]
- [tex]\( r = 0.04 \)[/tex]
- [tex]\( t = 4 \)[/tex]
Plug these into the function:
[tex]\[
246.4 = \rho e^{0.04 \times 4}
\][/tex]
3. Simplify the Exponential Expression: Calculate [tex]\( e^{0.04 \times 4} \)[/tex].
- First calculate [tex]\( 0.04 \times 4 = 0.16 \)[/tex].
- Then compute [tex]\( e^{0.16} \)[/tex].
4. Solve for [tex]\( \rho \)[/tex]: Now we need to express [tex]\( \rho \)[/tex] in terms of known values:
[tex]\[
\rho = \frac{246.4}{e^{0.16}}
\][/tex]
5. Numerical Evaluation: Substitute the computed value of [tex]\( e^{0.16} \)[/tex] to find [tex]\( \rho \)[/tex].
- Let's assume the calculation gives a result of approximately 209.97 for [tex]\( \rho \)[/tex].
6. Determine the Closest Option: Finally, compare this result to the multiple-choice answers:
- A. 50
- B. 289
- C. 1220
- D. 210
The number 209.97 is closest to option D. 210.
Therefore, the approximate value of [tex]\( P \)[/tex] is [tex]\( \boxed{210} \)[/tex].
Here's a step-by-step breakdown of how to solve it:
1. Understand the Function: The function provided is [tex]\( f(t) = \rho e^{rt} \)[/tex]. This is an exponential function where [tex]\( \rho \)[/tex] is the initial value or the coefficient, [tex]\( r \)[/tex] is the growth rate, and [tex]\( t \)[/tex] is the time.
2. Substitute the Known Values: We have:
- [tex]\( f(4) = 246.4 \)[/tex]
- [tex]\( r = 0.04 \)[/tex]
- [tex]\( t = 4 \)[/tex]
Plug these into the function:
[tex]\[
246.4 = \rho e^{0.04 \times 4}
\][/tex]
3. Simplify the Exponential Expression: Calculate [tex]\( e^{0.04 \times 4} \)[/tex].
- First calculate [tex]\( 0.04 \times 4 = 0.16 \)[/tex].
- Then compute [tex]\( e^{0.16} \)[/tex].
4. Solve for [tex]\( \rho \)[/tex]: Now we need to express [tex]\( \rho \)[/tex] in terms of known values:
[tex]\[
\rho = \frac{246.4}{e^{0.16}}
\][/tex]
5. Numerical Evaluation: Substitute the computed value of [tex]\( e^{0.16} \)[/tex] to find [tex]\( \rho \)[/tex].
- Let's assume the calculation gives a result of approximately 209.97 for [tex]\( \rho \)[/tex].
6. Determine the Closest Option: Finally, compare this result to the multiple-choice answers:
- A. 50
- B. 289
- C. 1220
- D. 210
The number 209.97 is closest to option D. 210.
Therefore, the approximate value of [tex]\( P \)[/tex] is [tex]\( \boxed{210} \)[/tex].
