The earth's atmospheric pressure, P, in terms of height above sea level is often modeled by an exponential decay function. The pressure at sea level is 1013 millibars and the pressure decreases by 14% for every kilometer above sea level.
(a) What is the atmospheric pressure at 50 km?
(b) Estimate the altitude h at which the atmospheric pressure equals 950 millibars.
To solve these problems, we will use the given exponential decay model. Let's denote the atmospheric pressure at height h as P(h). We know that the pressure at sea level is 1013 millibars, so P(0) = 1013.
The pressure decreases by 14% for every kilometer above sea level. This means that P(h) is 86% of P(h-1). In percentage form, this can be written as:
P(h) = 0.86 * P(h-1)
(a) To find the atmospheric pressure at 50 km, let's use the recursive formula:
P(50) = 0.86 * P(49)
P(49) = 0.86 * P(48)
...
P(1) = 0.86 * P(0)
Using this recursive formula, we can calculate the pressure step by step:
P(1) = 0.86 * 1013 = 870.98
P(2) = 0.86 * 870.98 ≈ 749.87
...
P(50) ≈ 0.86 * ... * 0.86 * 1013
Continuing this calculation, we can find that the atmospheric pressure at 50 km is approximately 9.77 millibars.
(b) To estimate the altitude h at which the atmospheric pressure equals 950 millibars, we need to solve the equation P(h) = 950.
0.86^h * 1013 = 950
Using logarithms, we can find the value of h:
log(0.86^h * 1013) = log(950)
h * log(0.86) + log(1013) = log(950)
h * log(0.86) = log(950) - log(1013)
h * log(0.86) ≈ -0.0251631
h ≈ -0.0251631 / log(0.86)
Using a calculator, we can find that h is approximately 16.036 km. Therefore, the estimated altitude at which the atmospheric pressure equals 950 millibars is about 16.036 km.
To find the atmospheric pressure at a certain altitude using an exponential decay function, we can use the formula:
P = P0 * e^(-r * h),
where P is the pressure at a given altitude, P0 is the pressure at sea level, e is Euler's constant (~2.71828), r is the decay rate (in this case, -0.14), and h is the altitude.
(a) To find the atmospheric pressure at 50 km, we can substitute the values into the formula:
P = 1013 * e^(-0.14 * 50).
Using a calculator, we can evaluate this expression:
P ≈ 1013 * e^(-7) ≈ 1013 * 0.000911 ≈ 0.926 ≈ 926 millibars.
Therefore, the atmospheric pressure at 50 km is approximately 926 millibars.
(b) To estimate the altitude h at which the atmospheric pressure equals 950 millibars, we need to solve the equation:
950 = 1013 * e^(-0.14 * h).
Divide both sides by 1013:
0.9397 ≈ e^(-0.14 * h).
Take the natural logarithm (ln) of both sides to isolate h:
ln(0.9397) ≈ -0.14 * h.
Now we can solve for h:
h ≈ ln(0.9397) / (-0.14).
Using a calculator again, we can compute this value:
h ≈ -0.257 / (-0.14) ≈ 1.836 km.
Therefore, the estimated altitude at which the atmospheric pressure equals 950 millibars is approximately 1.836 km.
.86=1e^k(1km)
so the drill is to find k
ln .86=k
k=-0.15082289
so at 50km
pressure= 1013e^(-.150*50)=.538millbars
check that.