Write a formula that relates temperature t in degrees Fahrenheit at altitude h (in thousands of feet) and a ground temperature of 65ºF. State any restrictions on h.
b. Find the temperature at 11,000 ft above Earth’s surface.
To relate temperature (t) in degrees Fahrenheit at altitude (h) to a ground temperature of 65ºF, you can use the following formula, which is based on the adiabatic lapse rate at an average value of 3.5ºF per 1,000 feet:
t = 65 - 3.5h
The restriction on h is that it should be specified in thousands of feet. So, a valid range for h would be any non-negative real number less than or equal to 11. Therefore, h must satisfy the condition 0 ≤ h ≤ 11.
To find the temperature at 11,000 ft above Earth's surface (h = 11), you can substitute the value of h into the formula:
t = 65 - 3.5 * 11
t = 65 - 38.5
t = 26.5ºF
So, the temperature at 11,000 ft above Earth's surface is 26.5ºF.
To relate the temperature t in degrees Fahrenheit at altitude h in thousands of feet to a ground temperature of 65ºF, we can use the following formula:
t = 65 - 1.98h
This formula is based on the Standard Atmosphere Model, which assumes a lapse rate of approximately 1.98 degrees Fahrenheit per thousand feet of altitude. The temperature decreases by 1.98 degrees Fahrenheit for every thousand feet increase in altitude.
Now, let's use this formula to find the temperature at 11,000 ft above Earth's surface:
t = 65 - 1.98h
t = 65 - 1.98(11)
t = 65 - 21.78
t ≈ 43.22ºF
Thus, the temperature at 11,000 ft above Earth's surface is approximately 43.22ºF.
Restrictions on h:
It's important to note that the given formula is an approximation based on the Standard Atmosphere Model. It assumes standard atmospheric conditions and may not be accurate in highly variable or specific situations, such as closer to the Earth's surface or in extreme weather conditions. Furthermore, this formula is primarily applicable in the tropospheric region within the Earth's atmosphere, up to an altitude of approximately 36,000 feet. Therefore, it may not be applicable to altitudes beyond this range or in the stratosphere and beyond.