Answer :
To find the approximate value of [tex]\( P \)[/tex] in the function [tex]\( f(t) = P \times e^{rt} \)[/tex], when given that [tex]\( f(5) = 288.9 \)[/tex] and [tex]\( r = 0.05 \)[/tex], follow these steps:
1. Write the function using given values:
The function can be expressed as [tex]\( f(t) = P \times e^{(r \times t)} \)[/tex].
We know that when [tex]\( t = 5 \)[/tex], [tex]\( f(5) = 288.9 \)[/tex].
2. Substitute the known values into the equation:
[tex]\[
288.9 = P \times e^{(0.05 \times 5)}
\][/tex]
3. Calculate the exponent:
First, find the value of [tex]\( 0.05 \times 5 \)[/tex]:
[tex]\[
0.05 \times 5 = 0.25
\][/tex]
4. Calculate [tex]\( e^{0.25} \)[/tex]:
The approximate value of [tex]\( e^{0.25} \)[/tex] is about 1.284.
5. Solve for [tex]\( P \)[/tex]:
Now, substitute the value of [tex]\( e^{0.25} \)[/tex] into the equation:
[tex]\[
288.9 = P \times 1.284
\][/tex]
6. Divide both sides by 1.284 to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{288.9}{1.284} \approx 225
\][/tex]
Thus, the approximate value of [tex]\( P \)[/tex] is 225. So, the correct answer is D. 225.
1. Write the function using given values:
The function can be expressed as [tex]\( f(t) = P \times e^{(r \times t)} \)[/tex].
We know that when [tex]\( t = 5 \)[/tex], [tex]\( f(5) = 288.9 \)[/tex].
2. Substitute the known values into the equation:
[tex]\[
288.9 = P \times e^{(0.05 \times 5)}
\][/tex]
3. Calculate the exponent:
First, find the value of [tex]\( 0.05 \times 5 \)[/tex]:
[tex]\[
0.05 \times 5 = 0.25
\][/tex]
4. Calculate [tex]\( e^{0.25} \)[/tex]:
The approximate value of [tex]\( e^{0.25} \)[/tex] is about 1.284.
5. Solve for [tex]\( P \)[/tex]:
Now, substitute the value of [tex]\( e^{0.25} \)[/tex] into the equation:
[tex]\[
288.9 = P \times 1.284
\][/tex]
6. Divide both sides by 1.284 to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{288.9}{1.284} \approx 225
\][/tex]
Thus, the approximate value of [tex]\( P \)[/tex] is 225. So, the correct answer is D. 225.
