High School

If [tex]$f(4)=246.4$[/tex] when [tex]$r=0.04$[/tex] for the function [tex]$f(t)=\rho e^t$[/tex], then what is the approximate value of [tex][tex]$\rho$[/tex][/tex]?

A. 289
B. 50
C. 1220
D. 210

Answer :

To solve the problem, we need to find the value of [tex]\( P \)[/tex] when given the function [tex]\( f(t) = P \cdot e^{rt} \)[/tex]. We know that [tex]\( f(4) = 246.4 \)[/tex] when [tex]\( r = 0.04 \)[/tex].

Step-by-step solution:

1. Write down the given function and information:

- We're given the function: [tex]\( f(t) = P \cdot e^{rt} \)[/tex].
- Given data: [tex]\( f(4) = 246.4 \)[/tex] and [tex]\( r = 0.04 \)[/tex].
- We need to find [tex]\( P \)[/tex].

2. Substitute the known values into the function:

- We substitute [tex]\( t = 4 \)[/tex] and [tex]\( r = 0.04 \)[/tex] into the function:
[tex]\[
f(4) = P \cdot e^{0.04 \times 4}
\][/tex]

3. Use the given [tex]\( f(4) \)[/tex]:

- We know [tex]\( f(4) = 246.4 \)[/tex], so:
[tex]\[
246.4 = P \cdot e^{0.16}
\][/tex]

4. Solve for [tex]\( P \)[/tex]:

- Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{246.4}{e^{0.16}}
\][/tex]

5. Calculate [tex]\( e^{0.16} \)[/tex]:

- Calculate the value of [tex]\( e^{0.16} \)[/tex] approximately to find [tex]\( P \)[/tex].

6. Find [tex]\( P \)[/tex] and round to the closest answer choice:

- The calculation yields [tex]\( P \approx 209.968 \)[/tex].
- The closest whole number and given answer choice is 210.

Hence, the approximate value of [tex]\( P \)[/tex] is [tex]\( \boxed{210} \)[/tex].