Question

Atmospheric Pressure. The atmosphere in the lower stratosphere
decreases exponentially from 473 lb/ft^{3} at 152 ft to 51
lb/ft^{2} 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/ft^{2} ?

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*b^{46193}............(ii)

Dividing (i) and (ii) we get,

473/51 = b^{-36000}/b^{46193}

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

i.e., b^{82193} = 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/ft^{2}.

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 .

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