Answer :
Let's solve this step-by-step to find the approximate value of [tex]\( P \)[/tex].
We are given the function [tex]\( f(t) = P e^t \)[/tex] and told that [tex]\( f(3) = 191.5 \)[/tex] when [tex]\( r = 0.03 \)[/tex].
Step 1: Understand that the expression given as [tex]\( f(t) = P e^t \)[/tex] involves an exponential function. Here, [tex]\( t \)[/tex] is our variable, and [tex]\( P \)[/tex] and [tex]\( r \)[/tex] are constants.
Step 2: Calculate the expression inside the exponent for time [tex]\( t = 3 \)[/tex]. The exponent becomes [tex]\( r \times t = 0.03 \times 3 \)[/tex].
Step 3: Substitute [tex]\( t = 3 \)[/tex] and [tex]\( r = 0.03 \)[/tex] into the equation. We need to find the value of [tex]\( P \)[/tex] by isolating it:
[tex]\[ 191.5 = P \times e^{0.03 \times 3} \][/tex]
[tex]\[ 191.5 = P \times e^{0.09} \][/tex]
Step 4: Calculate the value of [tex]\( e^{0.09} \)[/tex]. It is approximately [tex]\( 1.09417 \)[/tex].
Step 5: Solve for [tex]\( P \)[/tex] by dividing both sides of the equation by [tex]\( e^{0.09} \)[/tex]:
[tex]\[ P = \frac{191.5}{1.09417} \][/tex]
Step 6: Calculate the result:
[tex]\[ P \approx 175.0178 \][/tex]
The approximate value of [tex]\( P \)[/tex] is 175. So, the correct answer is:
D. 175
We are given the function [tex]\( f(t) = P e^t \)[/tex] and told that [tex]\( f(3) = 191.5 \)[/tex] when [tex]\( r = 0.03 \)[/tex].
Step 1: Understand that the expression given as [tex]\( f(t) = P e^t \)[/tex] involves an exponential function. Here, [tex]\( t \)[/tex] is our variable, and [tex]\( P \)[/tex] and [tex]\( r \)[/tex] are constants.
Step 2: Calculate the expression inside the exponent for time [tex]\( t = 3 \)[/tex]. The exponent becomes [tex]\( r \times t = 0.03 \times 3 \)[/tex].
Step 3: Substitute [tex]\( t = 3 \)[/tex] and [tex]\( r = 0.03 \)[/tex] into the equation. We need to find the value of [tex]\( P \)[/tex] by isolating it:
[tex]\[ 191.5 = P \times e^{0.03 \times 3} \][/tex]
[tex]\[ 191.5 = P \times e^{0.09} \][/tex]
Step 4: Calculate the value of [tex]\( e^{0.09} \)[/tex]. It is approximately [tex]\( 1.09417 \)[/tex].
Step 5: Solve for [tex]\( P \)[/tex] by dividing both sides of the equation by [tex]\( e^{0.09} \)[/tex]:
[tex]\[ P = \frac{191.5}{1.09417} \][/tex]
Step 6: Calculate the result:
[tex]\[ P \approx 175.0178 \][/tex]
The approximate value of [tex]\( P \)[/tex] is 175. So, the correct answer is:
D. 175
