Answer :
To find the approximate value of [tex]\( P \)[/tex] in the given exponential function [tex]\( f(i) = P \times e^{-r \times i} \)[/tex], we are provided with some values:
- [tex]\( f(3) = 191.5 \)[/tex]
- [tex]\( r = 0.03 \)[/tex]
- [tex]\( i = 3 \)[/tex]
The function can be expressed as:
[tex]\[ f(i) = P \times e^{-r \times i} \][/tex]
We need to find [tex]\( P \)[/tex] when [tex]\( f(3) = 191.5 \)[/tex]. This requires us to rearrange the equation to solve for [tex]\( P \)[/tex].
1. Start with the equation given:
[tex]\[ f(3) = P \times e^{-0.03 \times 3} \][/tex]
2. Substitute the known value of [tex]\( f(3) \)[/tex]:
[tex]\[ 191.5 = P \times e^{-0.09} \][/tex]
3. Calculate the value of [tex]\( e^{-0.09} \)[/tex]. Using a calculator, [tex]\( e^{-0.09} \approx 0.9139 \)[/tex].
4. Substitute this back into the equation:
[tex]\[ 191.5 = P \times 0.9139 \][/tex]
5. Now, solve for [tex]\( P \)[/tex] by dividing both sides by 0.9139:
[tex]\[ P = \frac{191.5}{0.9139} \][/tex]
6. Performing the division gives:
[tex]\[ P \approx 209.53 \][/tex]
Therefore, the approximate value of [tex]\( P \)[/tex] is closest to option C, which is 210.
- [tex]\( f(3) = 191.5 \)[/tex]
- [tex]\( r = 0.03 \)[/tex]
- [tex]\( i = 3 \)[/tex]
The function can be expressed as:
[tex]\[ f(i) = P \times e^{-r \times i} \][/tex]
We need to find [tex]\( P \)[/tex] when [tex]\( f(3) = 191.5 \)[/tex]. This requires us to rearrange the equation to solve for [tex]\( P \)[/tex].
1. Start with the equation given:
[tex]\[ f(3) = P \times e^{-0.03 \times 3} \][/tex]
2. Substitute the known value of [tex]\( f(3) \)[/tex]:
[tex]\[ 191.5 = P \times e^{-0.09} \][/tex]
3. Calculate the value of [tex]\( e^{-0.09} \)[/tex]. Using a calculator, [tex]\( e^{-0.09} \approx 0.9139 \)[/tex].
4. Substitute this back into the equation:
[tex]\[ 191.5 = P \times 0.9139 \][/tex]
5. Now, solve for [tex]\( P \)[/tex] by dividing both sides by 0.9139:
[tex]\[ P = \frac{191.5}{0.9139} \][/tex]
6. Performing the division gives:
[tex]\[ P \approx 209.53 \][/tex]
Therefore, the approximate value of [tex]\( P \)[/tex] is closest to option C, which is 210.
