Questions: The atmospheric pressure at a height of x meters above sea level is P(x) = 10^4 e^(-0.0012 x) kilograms per square meter. What is the rate of change of the pressure with respect to the height at x=400 meters? Round your answer to 2 decimal places.

The atmospheric pressure at a height of x meters above sea level is P(x) = 10^4 e^(-0.0012 x) kilograms per square meter. What is the rate of change of the pressure with respect to the height at x=400 meters? Round your answer to 2 decimal places.

Transcript text: Question The atmospheric pressure at a height of $x$ meters above sea level is $P(x)=10^{4} e^{-0.0012 x}$ kilograms per square meter. What is the rate of change of the pressure with respect to the height at $x=400$ meters? Round your answer to 2 decimal places. Provide your answer below: $\square$

Solution

Solution Steps

Step 1: Define the Pressure Function

The atmospheric pressure at a height of $ x $ meters above sea level is given by the function $ P(x) = 10^{4} e^{-0.0012 x} $ kg/m².

Step 2: Compute the Derivative

To find the rate of change of pressure with respect to height, we compute the derivative $ \frac{dP}{dx} $. The derivative is given by: \[ \frac{dP}{dx} = -0.0012 \cdot 10^{4} e^{-0.0012 x} \]

Step 3: Evaluate the Derivative at $ x = 400 $

We evaluate the derivative at $ x = 400 $: \[ \frac{dP}{dx} \bigg|_{x=400} = -0.0012 \cdot 10^{4} e^{-0.0012 \cdot 400} \] Calculating this gives us approximately $ -7.4254 $ kg/m³.