Q. 111 - Precalculus Enhanced with Graphing Utilities

Question

The relative humidity is the ratio (expressed as a percent) of the amount of water vapor in the air to the maximum amount that it can hold at a specific temperature. The relative humidity, R, is found using the following formula:

$R=10\frac{4221}{T+459.4}-\frac{4221}{D+459.4}+2$

where T is the air temperature (in $°F$) and D is the dew point temperature (in $°F$).

Part (a): Determine the relative humidity if the air temperature is $50°F$ and the dew point temperature is $41° F$.

Part (b): Determine the relative humidity if the air temperature is $68° F$ and the dew point temperature is $59° F$

Part (c): What is the relative humidity if the air temperature and the dew point temperature are the same?

Step-by-Step Solution

Answer

Part (a): The relative humidity if the air temperature is $50°F$ and the dew point temperature at $41°F$is $70.95\%$.

Part (b): The relative humidity if the air temperature is ${68}^{°}F$ and the dew point temperature at ${59}^{°}F$ is $72.61\%$.

Part (c): If the air and dew point temperature are the same, then the relative humidity is $100\%$.

Part (a) Step 1. Determine the relative humidity when T = 50 , D = 41 .

Consider the given function,

$$\begin{gathered} R=10\frac{4221}{T+459.4}-\frac{4221}{D+459.4}+2, \\ T=50, \\ D=41 \end{gathered}$$

Substitute the value in the given function,

$$\begin{gathered} R=10\frac{4221}{50+459.4}-\frac{4221}{41+459.4}+2 \\ =8.286-8.435+2 \\ =1.851 \end{gathered}$$

Replace the exponent with its evaluated value in the formula,

$$\begin{gathered} R={10}^{1.851} \\ =70.95 \end{gathered}$$

Part (b) Step 1. Determine the relative humidity when T = 68 , D = 59 .

Consider the given function,

$$\begin{gathered} R=10\frac{4221}{T+459.4}-\frac{4221}{D+459.4}+2, \\ T=68, \\ D=59 \end{gathered}$$

Substitute the value in the given function,

$$\begin{gathered} R=10\frac{4221}{68+459.4}-\frac{4221}{59+459.4}+2 \\ =8.003-8.142+2 \\ =1.861 \end{gathered}$$

Replace the exponent with its evaluated value in the formula,

$$\begin{gathered} R={10}^{1.861} \\ =72.61 \end{gathered}$$

Part (c) Step 1. If air temperature and dew point temperature are same.

Consider the given function,

$$\begin{gathered} R=10\frac{4221}{T+459.4}-\frac{4221}{D+459.4}+2, \\ T=10, \\ D=10 \end{gathered}$$

Substitute the values in the given function,

$$\begin{gathered} R=10\frac{4221}{10+459.4}-\frac{4221}{10+459.4}+2 \\ =\frac{4221}{469.4}-\frac{4221}{469.4}+2 \\ =2 \end{gathered}$$

Replace the exponent with its evaluated value in the formula,

$$\begin{gathered} R={10}^2 \\ =100 \end{gathered}$$