Question
Atmospheric pressure at sea level is about 1,013 millibars. The pressure decreases about 11% for each 1-kilometer increase in elevation. You can use a function to approximate the atmospheric pressure at an elevation of x kilometers. Is the function linear or exponential? Which equation represents the function
Answers
GPT-4o mini
To determine whether the function describing atmospheric pressure with respect to elevation is linear or exponential, we need to analyze how the pressure changes with elevation.
Given that atmospheric pressure at sea level is about 1,013 millibars, and the pressure decreases by about 11% for each 1-kilometer increase in elevation, this indicates a multiplicative decrease rather than a constant decrease. A linear function would imply a constant rate of change, while an exponential function describes a situation where the rate of change is proportional to the current value.
Since the pressure decreases by a percentage (11%) for each kilometer, we can express the atmospheric pressure \( P(x) \) at an elevation \( x \) kilometers with the following exponential decay function:
\[
P(x) = P_0 \cdot (1 - r)^x
\]
where:
- \( P_0 \) is the pressure at sea level (1,013 millibars),
- \( r \) is the rate of decrease (11% or 0.11),
- \( (1 - r) = 0.89 \) is the factor by which the pressure decreases each kilometer,
- \( x \) is the elevation in kilometers.
Plugging in the known values:
\[
P(x) = 1013 \cdot (0.89)^x
\]
Thus, the function is exponential rather than linear. The equation representing the function is:
\[
P(x) = 1013 \cdot (0.89)^x
\]
Given that atmospheric pressure at sea level is about 1,013 millibars, and the pressure decreases by about 11% for each 1-kilometer increase in elevation, this indicates a multiplicative decrease rather than a constant decrease. A linear function would imply a constant rate of change, while an exponential function describes a situation where the rate of change is proportional to the current value.
Since the pressure decreases by a percentage (11%) for each kilometer, we can express the atmospheric pressure \( P(x) \) at an elevation \( x \) kilometers with the following exponential decay function:
\[
P(x) = P_0 \cdot (1 - r)^x
\]
where:
- \( P_0 \) is the pressure at sea level (1,013 millibars),
- \( r \) is the rate of decrease (11% or 0.11),
- \( (1 - r) = 0.89 \) is the factor by which the pressure decreases each kilometer,
- \( x \) is the elevation in kilometers.
Plugging in the known values:
\[
P(x) = 1013 \cdot (0.89)^x
\]
Thus, the function is exponential rather than linear. The equation representing the function is:
\[
P(x) = 1013 \cdot (0.89)^x
\]