To find the value of [tex]\(f(7)\)[/tex] for the function [tex]\( f(t) = P e^{r t} \)[/tex] when [tex]\( P = 7 \)[/tex] and [tex]\( r = 0.07 \)[/tex], let's follow these steps:
1. Substitute the given values into the function:
[tex]\[
f(t) = 7 \cdot e^{0.07 t}
\][/tex]
We need to find [tex]\( f(7) \)[/tex]:
[tex]\[
f(7) = 7 \cdot e^{0.07 \cdot 7}
\][/tex]
2. Calculate the exponent:
[tex]\[
0.07 \cdot 7 = 0.49
\][/tex]
So the function becomes:
[tex]\[
f(7) = 7 \cdot e^{0.49}
\][/tex]
3. Evaluate the exponential term [tex]\( e^{0.49} \)[/tex]:
The value of [tex]\( e^{0.49} \)[/tex] is approximately 1.64971.
4. Multiply the result by 7:
[tex]\[
f(7) = 7 \cdot 1.64971 \approx 11.426213539687653
\][/tex]
5. Round the result to the nearest tenth:
The value [tex]\( 11.426213539687653 \)[/tex] rounded to the nearest tenth is 11.4.
So, the value of [tex]\( f(7) \)[/tex] to the nearest tenth is:
[tex]\[
\boxed{11.4}
\][/tex]
