Question

If the temperature is constant, then the atmospheric pressure P (in pounds per square inch) varies with the altitude above sea level h in accordance with the law

P = p0e−ᵏʰ

where

p₀

is the atmospheric pressure at sea level and k is a constant. If the atmospheric pressure is 14.7 lb/in² at sea level and 13 lb/in² at 4000 ft, find the atmospheric pressure at an altitude of 11,000 ft. (Round your answer to two decimal places.)

Answer 1

To find the atmospheric pressure at 11,000 ft, we determine the constant k using the given pressures at sea level and at 4000 ft, then apply it to the exponential model P = p0e−ᵅ6 to compute the pressure at 11,000 ft, rounding to two decimal places.

Step-by-step explanation:

To calculate the atmospheric pressure at an altitude of 11,000 ft using the given exponential model P = p0e−ᵅ6, we need to find the value of the constant k. We're given the atmospheric pressure at sea level (p0 = 14.7 lb/in²) and at an altitude of 4000 ft (P = 13 lb/in²). Plugging these values into our model, we get:

1. 13 = 14.7e−k(4000)
2. Divide both sides by 14.7: 13 / 14.7 = e−k(4000)
3. Take the natural logarithm of both sides: ln(13 / 14.7) = −k(4000)
4. Solve for k: k = −ln(13 / 14.7) / 4000

After finding the value of k, we use it to find the pressure at 11,000 ft:

1. Substitute k into the original model: P = 14.7e−k(11000)
2. Calculate P to find the atmospheric pressure at 11,000 ft and round to two decimal places.

Following through with these calculations will give us the atmospheric pressure at 11,000 ft.