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Answer :
To solve this problem, we need to find the value of [tex]\( P \)[/tex] for the function [tex]\( f(t) = P e^{rt} \)[/tex] given that [tex]\( f(3) = 191.5 \)[/tex] when [tex]\( r = 0.03 \)[/tex].
Here's a step-by-step solution:
1. Start with the given function:
[tex]\[ f(t) = P e^{rt} \][/tex]
2. Substitute the known values:
We know that when [tex]\( t = 3 \)[/tex], [tex]\( f(3) = 191.5 \)[/tex]. Also, the rate [tex]\( r = 0.03 \)[/tex].
[tex]\[ f(3) = 191.5 = P e^{0.03 \times 3} \][/tex]
3. Simplify the exponent:
Calculate the expression inside the exponent:
[tex]\[ 0.03 \times 3 = 0.09 \][/tex]
4. Calculate the exponential value:
Find the approximate value of [tex]\( e^{0.09} \)[/tex].
5. Solve for [tex]\( P \)[/tex]:
Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[ P = \frac{191.5}{e^{0.09}} \][/tex]
6. Calculate the value of [tex]\( P \)[/tex]:
Divide 191.5 by the calculated value of [tex]\( e^{0.09} \)[/tex]. The approximate result you get is [tex]\( 175.0178 \)[/tex].
Based on these calculations, the approximate value of [tex]\( P \)[/tex] is about 175. Therefore, the correct answer is:
D. 175
Here's a step-by-step solution:
1. Start with the given function:
[tex]\[ f(t) = P e^{rt} \][/tex]
2. Substitute the known values:
We know that when [tex]\( t = 3 \)[/tex], [tex]\( f(3) = 191.5 \)[/tex]. Also, the rate [tex]\( r = 0.03 \)[/tex].
[tex]\[ f(3) = 191.5 = P e^{0.03 \times 3} \][/tex]
3. Simplify the exponent:
Calculate the expression inside the exponent:
[tex]\[ 0.03 \times 3 = 0.09 \][/tex]
4. Calculate the exponential value:
Find the approximate value of [tex]\( e^{0.09} \)[/tex].
5. Solve for [tex]\( P \)[/tex]:
Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[ P = \frac{191.5}{e^{0.09}} \][/tex]
6. Calculate the value of [tex]\( P \)[/tex]:
Divide 191.5 by the calculated value of [tex]\( e^{0.09} \)[/tex]. The approximate result you get is [tex]\( 175.0178 \)[/tex].
Based on these calculations, the approximate value of [tex]\( P \)[/tex] is about 175. Therefore, the correct answer is:
D. 175
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