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
d)are the predictions made by the rule of thumb too large or too small?why?
please help me.....i can not solve it..i need these answer within 2 hours..please help me..
a) To find the equation of the tangent line at h=0, we need to find the derivative of the air pressure equation with respect to h and evaluate it at h=0.
Let's start by finding the derivative:
P = 30e^(-3.23x10^(-5)h)
Differentiating both sides of the equation with respect to h:
dP/dh = -3.23x10^(-5) * 30 * e^(-3.23x10^(-5)h)
Now, evaluate this derivative at h=0:
dP/dh = -3.23x10^(-5) * 30 * e^(-3.23x10^(-5)*0)
= -3.23x10^(-5) * 30 * 1 (since e^0 = 1)
= -0.000969
The derivative represents the slope of the tangent line at any point. So, the equation of the tangent line at h=0 is given by:
P - P0 = m(h - h0), where P0 is the air pressure at h=0 and m is the slope of the tangent line.
Substituting the values:
P - P0 = -0.000969(h - 0)
P - P0 = -0.000969h
b) The rule of thumb states that air pressure drops about 1 inch for every 1000-foot increase in height above sea level.
Let's assume the initial air pressure at sea level is P0. For every 1000-foot increase in height, the air pressure drops by 1 inch. Therefore, the formula for the air pressure given by this rule of thumb can be written as:
P = P0 - (h/1000)
c) The equation obtained in part (a) represents the exact equation for air pressure at a given altitude, which takes into account the exponential decay of air pressure with increasing altitude. On the other hand, the equation obtained in part (b) represents a simplified approximation based on a linear relationship between altitude and air pressure.
The rule of thumb works because it provides a rough estimate of the air pressure drop with altitude. It assumes a linear decrease in air pressure with increasing altitude, which holds true for moderate altitudes. However, as we go to higher altitudes, the exponential decay term in the exact equation becomes significant, and the linear approximation becomes less accurate.
d) The predictions made by the rule of thumb are generally too small. This is because the rule approximates the air pressure drop as a linear relationship, which ignores the exponential decay of air pressure with altitude. As a result, the predicted air pressure values will be lower than the actual values at higher altitudes.