Question

Atmospheric Pressure. The atmosphere in the lower stratosphere decreases exponentially from 473 lb/ft³ at 152 ft to 51 lb/ft² at 82,345 ft.

a) Find the exponnetial decay rate k, and write an equation for an equation for an exponential function that can be used to estimate the atmospheric preesure in the stratosphere h feet above 36,152 ft.

b) Estimate the atmospheric pressure at 50,000 ft (h = 50,000 - 36,152).

c) At what height is the atmosphere pressure 100 lb/ft² ?

d) What change in altitude will result in atmospheric pressure being halved ?

Answer 1

a) Let the exponential function be : y = a*b^h.

Given that y = 473 when h = (152-36152) = -36000 and y = 51 when h = (82345-36152) = 46193.

Then we have,

473 = a*b^-36000...........(i)

51 = a*b⁴⁶¹⁹³............(ii)

Dividing (i) and (ii) we get,

473/51 = b^-36000/b⁴⁶¹⁹³

i.e., b^-82193 = 473/51

i.e., b⁸²¹⁹³ = 51/473

i.e.,

i.e.,

Putting this in (i) we get,

473 = a*(0.9999729)^-36000

i.e., a = 473*(0.9999729)³⁶⁰⁰⁰

i.e.,

Therefore, the required exponential equation is : .

b) Given, h = 50000-36152 = 13848

Now,

i.e.,

Therefore, required atmospheric pressure is 122.52 lb/ft².

c) Given, y = 100.

Then,

i.e.,

i.e.,

i.e.,

Therefore, the required height is = (36000+21342.72) ft = 57342.72 ft.

d) By condition we have,

i.e.,

i.e.,

i.e.,

i.e.,

i.e.,

i.e.,

i.e.,

i.e.,

Therefore, the new altittude will be and the new altitude will decrease .