Chapter 12.2, Problem 63AYU

Cooling Air As a parcel of air rises (for example, as it is pushed over a mountain), it cools at the dry adiabatic lapse rate of ${5.5}^{\circ }F$ per 1000 feet until it reaches its dew point. If the ground temperature is ${67}^{\circ }F$, write a formula for the sequence of temperatures, $\left\{T_n\right\}$, of a parcel of air that has risen $n$ thousand feet. What is the temperature of a parcel of air if it has risen 5000 feet?

Source: National Aeronautics and Space Administration

To determine

To find: A parcel of air rises (for example, as it is pushed over a mountain), it cools at the dry adiabatic lapse rate of ${5.5}^{\circ }F$ per 1000 feet until it reaches its dew point. If the ground temperature is ${67}^{\circ }F$, write a formula for the sequence of temperatures, $\left\{T_n\right\}$, of a parcel of air that has risen $n$ thousand feet. What is the temperature of a parcel of air if it has risen 5000 feet?

Answer to Problem 63AYU

a. $T_n=67-5.5n$

Explanation of Solution

Given:

Lapse rate for every 1000 feet $=5.5°F$.

Ground Temperature $=67°F$.

Formula used:

The $n$th term of an arithmetic sequence can be found by the formula,

$a_n=a_0+nd$

Calculation:

a. To write the formula for sequence of temperature $\left\{T_n\right\}$:

From the data, we get $T_0=67$ and $d=-5.5$.

The sequence $T_n=T_0+nd=67-5.5n$, where $n$ is $n$th 1000 feet.

To determine

To find: A parcel of air rises (for example, as it is pushed over a mountain), it cools at the dry adiabatic lapse rate of ${5.5}^{\circ }F$ per 1000 feet until it reaches its dew point. If the ground temperature is ${67}^{\circ }F$, write a formula for the sequence of temperatures, $\left\{T_n\right\}$, of a parcel of air that has risen $n$ thousand feet. What is the temperature of a parcel of air if it has risen 5000 feet?

Answer to Problem 63AYU

b. $T_5=39.5°F$

Explanation of Solution

Given:

Lapse rate for every 1000 feet $=5.5°F$.

Ground Temperature $=67°F$.

Formula used:

The $n$th term of an arithmetic sequence can be found by the formula,

$a_n=a_0+nd$

Calculation:

b. To find the temperature of air at 5000 feet:

For 5000 ft, $n=5$

$T_5=67+5\left(-5.5\right)=67-27.7$

$T_5=39.5°F$