The relative humidity is the ratio (expressed as a percent)...

The relative humidity is the ratio (expressed as a percent) of the amount of water vapor in the air to the maximum amount that the air can hold at a specific temperature. The relative humidity, $R$, is found using the following formula: $$ R=10^{\left(\frac{4221}{T+459.4}-\frac{4221}{D+459.4}+2\right)} $$ where $T$ is the air temperature (in ${ }^{\circ} \mathrm{F}$ ) and $D$ is the dew point temperature (in $\left.^{\circ} \mathrm{F}\right)$ (a) Determine the relative humidity if the air temperature is $50^{\circ}$ Fahrenheit and the dew point temperature is $41^{\circ}$ Fahrenheit. (b) Determine the relative humidity if the air temperature is $68^{\circ}$ Fahrenheit and the dew point temperature is $59^{\circ}$ Fahrenheit. (c) What is the relative humidity if the air temperature and the dew point temperature are the same?

The relative humidity is the ratio (expressed as a percent) of the amount of water vapor in the air to the maximum amount that the air can hold at a specific temperature. The relative humidity, $R$, is found using the following formula:
$$
R=10^{\left(\frac{4221}{T+459.4}-\frac{4221}{D+459.4}+2\right)}
$$
where $T$ is the air temperature (in ${ }^{\circ} \mathrm{F}$ ) and $D$ is the dew point temperature (in $\left.^{\circ} \mathrm{F}\right)$
(a) Determine the relative humidity if the air temperature is $50^{\circ}$ Fahrenheit and the dew point temperature is $41^{\circ}$ Fahrenheit.
(b) Determine the relative humidity if the air temperature is $68^{\circ}$ Fahrenheit and the dew point temperature is $59^{\circ}$ Fahrenheit.
(c) What is the relative humidity if the air temperature and the dew point temperature are the same?

Key Concepts

Relative Humidity

Relative humidity is a measure of the amount of water vapor present in air, expressed as a percentage relative to the maximum amount of water vapor the air can hold at a particular temperature. It indicates how close the air is to being saturated and is important in understanding weather conditions, comfort levels, and potential for precipitation.

Dew Point

The dew point is the temperature at which air becomes saturated with moisture and water vapor begins to condense into liquid water, forming dew. It is a direct indicator of the moisture content in the air and is critical for calculating relative humidity, as it represents the point at which condensation occurs.

Exponential Relationships in Meteorological Formulas

Exponential functions are used in meteorological formulas to model the non-linear relationships between temperature and moisture capacity in the air. They allow for a more accurate representation of how air temperature influences the maximum amount of water vapor the air can hold, which is essential for calculating relative humidity.

Transcript

00:01 Okay, so we have this function here, r, which is equal to 10 raised to this fairly complex power.

00:08 And so what we're going to do is we're going to find the value of r when t is equal to 50 and when d is equal to 41.

00:20 So t is equal to 50, d is equal to 41.

00:24 So then r would be equal to 10 to the 4 ,221 divided by 50 plus, and i'll actually simplify this now, 50 plus 459 .4 is 509 .4.

00:42 And then we're minusing 4 ,221 divided by, oh, divided by, um, d plus 4 ,000 or plus 459 .4, which is equal to 41 plus 421, 459 .4 or 500 .4.

01:02 And then we're adding two.

01:04 So if we do this on a calculator, 4 ,221 divided by 509.

01:11 0 .4 minus 4 ,221 divided by 500 .4 is equal to negative .149.

01:22 So if we add two to that, we get 1 .2 .7.

01:27 8 .5.

01:29 So this is equal to 10 to the 1 .85 power, which is equal to, i plug this into a calculator, 70 .95.

01:41 So this is equal to 70 .95.

01:49 For part b, we're going to figure out what r is equal to when d is equal to 59 and when t is equal to 68.

02:02 So this would be, to the 4211 divided by 68 plus 459 .4 minus 4 ,221 divided by 59 .4.

02:22 And then we add 2 .4.

02:24 And so 68 plus 459 .4.

02:28 And so 68 plus 459 .4 is equal to 527.

02:33 Point four, so we have 10 raised to the 4 ,221st divided by 527 .4 minus 4221 divided by 459 .4 plus 59 is equal to 518 .4.

02:54 And then we're adding two...

Please give Ace some feedback