Atmospheric pressure \( P \) in pounds per square inch is represented by the formula \( P=14.7 e^{-0.21 x} \), where \( x \) is the number of miles above sea level. To the nearest foot, how high is the peak of a mountain with an atmospheric pressure of 8.562 pounds per square inch? (Hint: there are 5,280 feet in a mile) The mountain is high.

To find the height of the mountain where the atmospheric pressure \( P \) is 8.562 pounds per square inch, we'll use the given formula: \[ P = 14.7 e^{-0.21x} \] where \( x \) is the height in miles above sea level. ### Step 1: Solve for \( x \) Set \( P = 8.562 \) and solve for \( x \): \[ 8.562 = 14.7 e^{-0.21x} \] Divide both sides by 14.7: \[ \frac{8.562}{14.7} = e^{-0.21x} \Rightarrow 0.5823 \approx e^{-0.21x} \] Take the natural logarithm of both sides: \[ \ln(0.5823) = -0.21x \Rightarrow x = \frac{-\ln(0.5823)}{0.21} \] Calculate the natural logarithm: \[ \ln(0.5823) \approx -0.5405 \] So, \[ x \approx \frac{0.5405}{0.21} \approx 2.573 \text{ miles} \] ### Step 2: Convert Miles to Feet There are 5,280 feet in a mile. Multiply the height in miles by this conversion factor: \[ 2.573 \text{ miles} \times 5,280 \text{ feet/mile} \approx 13,586 \text{ feet} \] ### Conclusion The peak of the mountain is approximately **13,586 feet** above sea level. **Answer:** To the nearest foot, the mountain is 13 586 feet high.