Answer :
To solve the problem and find the approximate value of [tex]\( P \)[/tex], follow these steps:
1. Understand the function and given values:
- The function is [tex]\( f(t) = P \cdot e^t \)[/tex].
- We know [tex]\( f(3) = 191.5 \)[/tex] when [tex]\( r = 0.03 \)[/tex].
2. Substitute the known values:
- We substitute [tex]\( t = 3 \)[/tex] into the equation:
[tex]\[
191.5 = P \cdot e^{0.03 \times 3}
\][/tex]
3. Calculate the exponent:
- Compute the exponent:
[tex]\[
e^{0.03 \times 3} = e^{0.09}
\][/tex]
4. Solve for [tex]\( P \)[/tex]:
- Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{191.5}{e^{0.09}}
\][/tex]
5. Approximate the value of [tex]\( P \)[/tex]:
- Calculate this using the approximated value of [tex]\( e^{0.09} \)[/tex] and the division:
- With calculations (not shown here), you find that [tex]\( P \approx 175 \)[/tex].
6. Choose the nearest answer:
- Among the given options: A. 175, B. 471, C. 210, D. 78, the closest value to our result is 175.
Thus, the approximate value of [tex]\( P \)[/tex] is 175.
1. Understand the function and given values:
- The function is [tex]\( f(t) = P \cdot e^t \)[/tex].
- We know [tex]\( f(3) = 191.5 \)[/tex] when [tex]\( r = 0.03 \)[/tex].
2. Substitute the known values:
- We substitute [tex]\( t = 3 \)[/tex] into the equation:
[tex]\[
191.5 = P \cdot e^{0.03 \times 3}
\][/tex]
3. Calculate the exponent:
- Compute the exponent:
[tex]\[
e^{0.03 \times 3} = e^{0.09}
\][/tex]
4. Solve for [tex]\( P \)[/tex]:
- Rearrange the equation to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{191.5}{e^{0.09}}
\][/tex]
5. Approximate the value of [tex]\( P \)[/tex]:
- Calculate this using the approximated value of [tex]\( e^{0.09} \)[/tex] and the division:
- With calculations (not shown here), you find that [tex]\( P \approx 175 \)[/tex].
6. Choose the nearest answer:
- Among the given options: A. 175, B. 471, C. 210, D. 78, the closest value to our result is 175.
Thus, the approximate value of [tex]\( P \)[/tex] is 175.
