Problem 74 - Precalculus: Functions and Graphs

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 \(80.42^{\circ} \mathrm{F}\).

Step 1: Identify Given Values (Part A)

Identify the given temperature and dew point. We have:\(T = 70^{\circ} \mathrm{F}\) \(D = 55^{\circ} \mathrm{F}\)

Step 2: Substitute Values into Formula (Part A)

Substitute the values for \(T\) and \(D\) into the formula \(h = 227(T-D)\):\[h = 227(70 - 55)\]

Step 3: Calculate Height (Part A)

Perform the subtraction and multiply by 227 to find the height of the cloud base:\[h = 227 \times 15 = 3405 \text{ feet}\]

Step 4: Identify Given Values (Part B)

Identify the given dew point and cloud base height. We have:\(D = 65^{\circ} \mathrm{F}\) \(h = 3500 \text{ feet}\)

Step 5: Rearrange Formula to Solve for Temperature (Part B)

Rearrange \(h = 227(T-D)\) to solve for \(T\):\[T = \frac{h}{227} + D\]

Step 6: Substitute Values and Calculate Temperature (Part B)

Substitute the given values into the rearranged formula:\[T = \frac{3500}{227} + 65\] Calculate the division and add 65:\[T \approx 15.42 + 65 = 80.42^{\circ} \mathrm{F}\]

Key Concepts

Temperature Conversion

Temperature conversion is a fundamental part of understanding how different scales measure temperature. In precalculus, we're often concerned with converting between Fahrenheit and Celsius. To convert a temperature from Fahrenheit (\(^{\circ} F\),) to Celsius (\(^{\circ} C\),), you use the formula:

• \[ C = \frac{5}{9} (F - 32) \]

This conversion is essential to understanding problems that involve temperature in various contexts, like weather predictions or scientific experiments. In our exercise, the temperatures are already given in Fahrenheit, so there is no need to convert, but it's a good skill to have in mind. Vice-versa, to go from Celsius to Fahrenheit, the formula is:

• \[ F = \frac{9}{5} C + 32 \]

Mathematical Formulas

Mathematical formulas form the bedrock of problem-solving in mathematics. In this exercise, we use the formula for estimating the height of the cloud base:

• \[ h = 227(T - D) \]

Where:

• \(h\) is the height of the cloud base in feet.
• \(T\) is the ground temperature in degrees Fahrenheit.
• \(D\) is the dew point in degrees Fahrenheit.

This formula is straightforward. It requires substituting known values to calculate an unknown, whether it be the cloud base height or the ground temperature. Understanding how to manipulate and rearrange mathematical formulas is crucial, as seen in the second part of the exercise where the formula is solved for \(T\). Rearranging formulas involves basic algebraic skills, like isolating the variable of interest by performing inverse operations.

Problem Solving

Problem-solving is a critical skill that applies the mathematical formulas and concepts learned to real-world situations. In this exercise, the problem-solving process involves:

• Clearly identifying what is given and what needs to be found.
• Choosing the appropriate formula based on the information provided.
• Substituting known values into the formula.
• Performing calculations step-by-step.
• Double-checking the calculations for accuracy.

It's essential to break down the problem into smaller, manageable parts, which is why identifying given values was one of the first steps in both parts. This approach not only simplifies complex problems but also reduces the likelihood of errors. Moreover, having a structured methodology makes it easier to solve similar future problems efficiently.

Height Calculation

Calculating the height, specifically in meteorological contexts like cloud base height, entails understanding how changes in temperature affect atmospheric conditions. The exercise uses a specific formula to estimate the cloud base height:

• \[h = 227(T - D) \]

The calculation involves the change in temperature, represented by the difference \((T - D)\), which influences how high the moisture in the ground air can rise before cooling to the dew point and forming clouds.
This exercise illustrates the direct linear relationship between the temperature difference and height change: the bigger the difference between ground temperature and dew point, the higher the cloud base. Such calculations are significant in weather forecasts and understanding atmospheric phenomena. They help meteorologists predict weather patterns and advise on flying conditions, agriculture planning, and other activities affected by cloud cover.