To solve for the air pressure at an altitude of 3,000 meters using the given function [tex]\( P(h) = P_0 \cdot e^{-0.00012 h} \)[/tex]:
1. Identifying Given Values:
- [tex]\( h = 3000 \)[/tex] meters
- The function formula is [tex]\( P(h) = P_0 \cdot e^{-0.00012 h} \)[/tex].
2. Substituting the Height into the Formula:
Substitute [tex]\( h \)[/tex] with 3000:
[tex]\[
P(3000) = P_0 \cdot e^{-0.00012 \cdot 3000}
\][/tex]
3. Simplifying the Exponent:
Calculate the exponent:
[tex]\[
-0.00012 \cdot 3000 = -0.36
\][/tex]
4. Rewriting the Expression:
Replace the exponent in the equation:
[tex]\[
P(3000) = P_0 \cdot e^{-0.36}
\][/tex]
5. Final Expression:
Therefore, the air pressure at 3,000 meters above the Earth's surface can be represented by the expression:
[tex]\[
P_0 \cdot e^{-0.36}
\][/tex]
Thus, among the given options, the expression that best represents the air pressure at an altitude of 3,000 meters is:
D. [tex]\( P_0 \cdot e^{-0.36} \)[/tex]
To verify, calculating the numerical result of this expression yields roughly 0.697676326071031 when [tex]\( P_0 \)[/tex] is assumed to be 1, which is consistent with our understanding.
