Problem 38 - Algebra and Trigonometry with Analytic Geometry

Question

The height \(h\) (in feet) of the cloud base can be estimated using \(h=227(T-D)\), where \(T\) is the ground temperature and \(D\) is the dew point. (a) If the temperature is \(70^{\circ} \mathrm{F}\) and the dew point is \(55^{\circ} \mathrm{F}\), find the height of the cloud base. (b) If the dew point is \(65^{\circ} \mathrm{F}\) and the cloud base is 3500 feet, estimate the ground temperature.

Step-by-Step Solution

Answer

(a) 3405 feet; (b) approximately 80.42°F.

Step 1: Identify Given Values for Part (a)

For part (a), the given values are the ground temperature \(T = 70^{\circ} \mathrm{F}\) and the dew point \(D = 55^{\circ} \mathrm{F}\). We need to find the height \(h\) of the cloud base using the formula \(h = 227(T - D)\).

Step 2: Calculate Cloud Base Height for Part (a)

Using the formula \(h = 227(T - D)\), substitute the known values: \(T = 70\) and \(D = 55\). Calculate: \[ h = 227(70 - 55) = 227 \times 15 = 3405\]Therefore, the height of the cloud base is 3405 feet.

Step 3: Identify Given Values for Part (b)

For part (b), the dew point \(D = 65^{\circ} \mathrm{F}\) and the cloud base height \(h = 3500\) feet. We need to estimate the ground temperature \(T\) using the formula \(h = 227(T - D)\).

Step 4: Rearrange and Solve for Ground Temperature in Part (b)

First, rearrange the formula \(h = 227(T - D)\) to solve for \(T\): \[ T = \frac{h}{227} + D \]Substitute the known values: \(h = 3500\) and \(D = 65\). Calculate:\[ T = \frac{3500}{227} + 65 \approx 15.42 + 65 = 80.42\]The estimated ground temperature is approximately \(80.42^{\circ} \mathrm{F}\).

Key Concepts

Temperature Estimation

Temperature estimation is a key aspect in many weather-related calculations, including the determination of cloud base height. Understanding how changes in temperature affect measurements is crucial. In our exercise, we used the formula related to cloud base height where temperature plays a major role.

Consider this: if you know the cloud base height and the dew point, you can rearrange the formula to estimate the current ground temperature. This is called "back-calculation." It means you're working backwards from known values to find an unknown, like the ground temperature in our example.

To solve for temperature, we rearranged the formula:

• Start with: \( h = 227(T - D) \)
• Solve for \(T\): \( T = \frac{h}{227} + D \)

Substituting known values can then easily help estimate the temperature.

Dew Point

The dew point is a critical factor in weather and climate studies. It helps to gauge moisture in the air. In simple terms, it's the temperature at which air becomes saturated with moisture and water droplets start to form. Understanding the dew point helps meteorologists predict fog, dew, or frost, and in our exercise, it was used to find cloud base height.

Here's how it works: the difference between the ground temperature and the dew point can be used to estimate other weather conditions, like cloud base height. In our exercise, we found this difference to be crucial.

When the value of the dew point is close to the temperature, the air is humid. The difference, \(T - D\), in our formula \(h = 227(T - D)\), helps determine how high above the ground the clouds begin. A smaller difference means the base of the clouds will be lower, indicating moist air.

Cloud Base Height

Estimating the cloud base height is an essential part of predicting weather conditions. It tells us how high above the ground clouds begin to form, which can impact visibility and precipitation. In meteorology, this calculation is fundamental for weather forecasts.

Using the formula \( h = 227(T - D) \), we can estimate this height when given the temperature and dew point. Let's break it down:

• \(T\) is the ground temperature.
• \(D\) is the dew point.
• Subtract \(D\) from \(T\) to find the temperature difference.
• Multiply this difference by 227 to find \(h\), the cloud base height.

This calculation provides an approximate height because atmospheric conditions can vary, but it offers a good estimation of where clouds will start to form. It's especially useful for pilots and meteorologists planning around weather patterns.