Air pressure at sea level is 30 inches of mercury. At an altitude of h feet above the, air pressure, P, in inches of mercury, is given by,
P =30e^(-3.23x10-5h)
a)Find th equation of the tangent line at h=0.
b)A rule of thumb is given by travelers is that air pressure drops about 1 inch for every 1000-foot increase in height above sea level. Write a formula for the air pressure given by this rule of thumb
c)what is the relation between your answer to part (a) (b).explain why the rule of thumb works
a) To find the equation of the tangent line at h=0, we need to first find the derivative of the function P with respect to h.
P(h) = 30e^(-3.23x10^(-5)h)
Using the chain rule, the derivative of P(h) is:
P'(h) = (-3.23x10^(-5)) * 30 * e^(-3.23x10^(-5)h)
Now, we can substitute h=0 into P'(h) to find the slope of the tangent line at h=0:
P'(0) = (-3.23x10^(-5)) * 30 * e^(-3.23x10^(-5)(0))
P'(0) = (-3.23x10^(-5)) * 30 * e^(0)
P'(0) = (-3.23x10^(-5)) * 30 * 1
P'(0) = -0.000969
This gives us the slope of the tangent line. Now, since h=0 is the point of tangency, we can use the point-slope form equation of a line to find the equation of the tangent line:
y - y1 = m(x - x1)
Substituting the values, we get:
y - P(0) = -0.000969(x - 0)
Since P(0) = 30 (given in the problem), we can simplify further:
y - 30 = -0.000969x
Therefore, the equation of the tangent line at h=0 is:
P = -0.000969x + 30
b) According to the rule of thumb, air pressure drops about 1 inch for every 1000-foot increase in height above sea level. This means that the change in air pressure is directly proportional to the change in height.
Let's assume that P0 represents the sea-level air pressure, P represents the air pressure at a given height above sea level, and h represents the height above sea level in feet.
According to the rule of thumb, for every 1000-foot increase in height, the air pressure drops by 1 inch. Therefore, we can set up a proportionality equation:
(P0 - P) / 1 = h / 1000
Simplifying and rearranging the equation, we get:
P = P0 - (h / 1000)
This formula gives the air pressure, P, at a given height, h, above sea level based on the rule of thumb.
c) The relation between the answer to part (a) and part (b) is that the slope of the tangent line at h=0 in part (a) (-0.000969) represents the rate of change of air pressure with respect to height at sea level. This is consistent with the rule of thumb in part (b) where for every 1000-foot increase in height, the air pressure drops by 1 inch.
The rule of thumb approximates the exponential decay relationship between air pressure and height with a simpler linear relationship. This linear approximation holds reasonably well for relatively small changes in height near sea level, where the exponential decay is not significant. As we move further from sea level, the linear approximation becomes less accurate.
a) To find the equation of the tangent line at h=0, we need to find the slope of the tangent line.
We can do this by taking the derivative of the function P with respect to h:
P = 30e^(-3.23x10^(-5)h)
dP/dh = -3.23x10^(-5) * 30e^(-3.23x10^(-5)h)
Now, substitute h=0 into the derivative:
dP/dh = -3.23x10^(-5) * 30e^(-3.23x10^(-5)(0))
= -3.23x10^(-5) * 30e^0
= -3.23x10^(-5) * 30
So, the slope of the tangent line at h=0 is -3.23x10^(-5) * 30.
The equation of a straight line can be written as y = mx + c, where m is the slope and c is the y-intercept. Since the tangent line passes through h=0, the y-intercept will be the value of P at h=0.
Substituting h=0 into the original equation of P:
P = 30e^(-3.23x10^(-5)h)
P = 30e^(-3.23x10^(-5)(0))
P = 30e^0
P = 30
So, the y-intercept is 30.
Therefore, the equation of the tangent line at h=0 is P = (-3.23x10^(-5) * 30)h + 30.
b) The rule of thumb states that air pressure drops about 1 inch for every 1000-foot increase in height above sea level. This can be expressed as P = 30 - (h/1000), where P is the air pressure in inches of mercury and h is the height in feet.
c) The equation derived in part (a) is an exact mathematical representation of how air pressure changes with altitude. It is based on the physics and behavior of the atmosphere.
The rule of thumb described in part (b) is a simplified approximation of the mathematical equation. It is a convenient way for travelers to estimate air pressure changes during altitude changes. While it may not be as accurate as the mathematical equation, it provides a quick and easy rule to estimate air pressure changes.