Problem 73 - Precalculus: Functions and Graphs

Question

Below the cloud base, the air temperature \(T\left(\text { in }^{\circ} \mathrm{F}\right)\) at height \(h\) (in feet) can be approximated by the equation \(T=T_{0}-\left(\frac{5.5}{1000}\right) h,\) where \(T_{0}\) is the temperature at ground level. (a) Determine the air temperature at a height of 1 mile if the ground temperature is \(70^{\circ} \mathrm{F}\). (b) At what altitude is the temperature freezing?

Step-by-Step Solution

Answer

(a) The temperature at 1 mile is approximately 40.96°F. (b) Temperature is freezing at approximately 6909 feet.

Step 1: Convert miles to feet

To solve part (a), we first need to convert the height from miles to feet. We know that 1 mile is equal to 5280 feet. Therefore, at a height of 1 mile, the height \(h\) in feet is \(5280\) feet.

Step 2: Substitute values into the temperature equation

Using the given equation \(T = T_0 - \left(\frac{5.5}{1000}\right) h\), substitute \(T_0 = 70\) and \(h = 5280\) feet to find the air temperature at 1 mile. Calculate as follows:\[T = 70 - \left(\frac{5.5}{1000}\right) \times 5280\]

Step 3: Evaluate the expression

Calculate the expression \(\left(\frac{5.5}{1000}\right) \times 5280\) as follows:\[\frac{5.5}{1000} \times 5280 = 29.04\]Subtract this value from 70 to get:\[T = 70 - 29.04 = 40.96\]The air temperature at a height of 1 mile is approximately \(40.96^{\circ}F\).

Step 4: Determine altitude for freezing temperature

For part (b), we need the altitude \(h\) where the temperature is \(32^{\circ}F\). Set \(T = 32\) in the equation:\[32 = 70 - \left(\frac{5.5}{1000}\right) h\]

Step 5: Solve for altitude

Rearrange the equation from Step 4 to solve for \(h\):\[\left(\frac{5.5}{1000}\right) h = 70 - 32 = 38\]Then find \(h\) by solving:\[h = \frac{38 \times 1000}{5.5}\]

Step 6: Calculate the result

Calculate \(h\) from Step 5:\[h = \frac{38000}{5.5} \approx 6909.09\]Thus, the altitude at which the temperature is freezing is approximately 6909 feet.

Key Concepts

Understanding Temperature Gradient

The term "temperature gradient" refers to how temperature changes with distance. In this problem, temperature falls as we go higher in altitude. This is specific to a rate of \(-\frac{5.5}{1000}\) degree Fahrenheit per foot. Essentially, it measures how rapidly the temperature decreases as altitude increases.

• The steeper the gradient, the quicker the temperature drops with height.
• In this exercise, it means a reduction of 5.5 degrees Fahrenheit for every 1000 feet.

Grasping this concept helps in predicting air temperature through simple calculations. Knowing how temperature changes with height can be crucial in aviation, meteorology, and mountain and road construction.

Converting Altitude Measurements

Altitude conversion is crucial when dealing with different units of measurement. For example, altitude is often measured in feet in the U.S., but sometimes presented in miles. To ensure consistency, conversion is necessary.

• 1 mile equals 5280 feet.
• To convert any number of miles to feet, multiply by 5280.

This conversion is needed in our exercise to use the temperature equation properly, which runs on feet rather than miles. For instance, converting 1 mile to 5280 feet is a first step before plugging values into the equation.

Applying the Temperature Equation

The temperature equation used in this exercise is a formula that estimates air temperature at a given altitude. It is represented as \(T = T_0 - \left(\frac{5.5}{1000}\right)h\). Here,

• \(T\) is the temperature at altitude \(h\).
• \(T_0\) is the ground temperature.
• \(h\) is the height in feet.

This linear equation shows a direct relationship between altitude and temperature. To find the temperature, substitute the known values for \(T_0\) and \(h\), and solve the formula step by step. In our case, this helps calculate the air temperature at 5280 feet from a known ground temperature.

Finding the Freezing Point Altitude

When solving for the altitude where the temperature reaches the freezing point, set \(T\) at \(32^{\circ}F\), which is the freezing point of water.

• Rearrange the temperature equation to solve for \(h\).
• Plug in the values for \(T\) and \(T_0\).

Solving this provides the altitude where the temperature equals \(32^{\circ}F\). In this exercise, we rearranged the equation to isolate \(h\), calculated the value on one side, and found that the altitude where the temperature is freezing is around 6909 feet. Understanding how to reach this altitude aids in various fields including forecasting snow levels and assessing freezing conditions.