Answer :
We are given the function
[tex]$$
f(t) = P e^{rt}
$$[/tex]
with the values [tex]$f(4) = 246.4$[/tex] and [tex]$r = 0.04$[/tex]. We want to solve for [tex]$P$[/tex].
Step 1: Substitute the given values into the function at [tex]$t = 4$[/tex].
[tex]$$
246.4 = P e^{0.04 \times 4}
$$[/tex]
Step 2: Simplify the exponent.
[tex]$$
0.04 \times 4 = 0.16
$$[/tex]
Thus, the equation becomes
[tex]$$
246.4 = P e^{0.16}
$$[/tex]
Step 3: Solve for [tex]$P$[/tex] by dividing both sides by [tex]$e^{0.16}$[/tex].
[tex]$$
P = \frac{246.4}{e^{0.16}}
$$[/tex]
Step 4: Evaluate [tex]$e^{0.16}$[/tex]. It is known that
[tex]$$
e^{0.16} \approx 1.17351
$$[/tex]
Step 5: Finally, divide to determine [tex]$P$[/tex].
[tex]$$
P \approx \frac{246.4}{1.17351} \approx 210
$$[/tex]
Thus, the approximate value of [tex]$P$[/tex] is [tex]$\boxed{210}$[/tex], which corresponds to option D.
[tex]$$
f(t) = P e^{rt}
$$[/tex]
with the values [tex]$f(4) = 246.4$[/tex] and [tex]$r = 0.04$[/tex]. We want to solve for [tex]$P$[/tex].
Step 1: Substitute the given values into the function at [tex]$t = 4$[/tex].
[tex]$$
246.4 = P e^{0.04 \times 4}
$$[/tex]
Step 2: Simplify the exponent.
[tex]$$
0.04 \times 4 = 0.16
$$[/tex]
Thus, the equation becomes
[tex]$$
246.4 = P e^{0.16}
$$[/tex]
Step 3: Solve for [tex]$P$[/tex] by dividing both sides by [tex]$e^{0.16}$[/tex].
[tex]$$
P = \frac{246.4}{e^{0.16}}
$$[/tex]
Step 4: Evaluate [tex]$e^{0.16}$[/tex]. It is known that
[tex]$$
e^{0.16} \approx 1.17351
$$[/tex]
Step 5: Finally, divide to determine [tex]$P$[/tex].
[tex]$$
P \approx \frac{246.4}{1.17351} \approx 210
$$[/tex]
Thus, the approximate value of [tex]$P$[/tex] is [tex]$\boxed{210}$[/tex], which corresponds to option D.
