Problem 20 - Algebra 2

Question

Suppose the temperature \(T\) in \(^{\circ} \mathrm{F}\) above the Earth's surface is given by \(T(h)=-3.6 h+68,\) where \(h\) is the height (in thousands of feet). Find the temperature at a height of \(10,000\) feet.

Step-by-Step Solution

Answer

The temperature at 10,000 feet is \( 32^{\circ} \mathrm{F} \).

Step 1: Convert Height to Function Units

First, notice that height \( h \) is given in thousands of feet. Since we need the height \( h \) in the formula, convert \( 10,000 \) feet to thousands of feet: \( h = \frac{10,000}{1,000} = 10 \).

Step 2: Plug the Height into the Temperature Equation

Now that we have \( h = 10 \), substitute it into the temperature function \( T(h) = -3.6h + 68 \). This gives us \( T(10) = -3.6(10) + 68 \).

Step 3: Simplify the Expression

Calculate \( -3.6 \times 10 \), which is \( -36 \). This simplifies the expression to \( T(10) = -36 + 68 \).

Step 4: Compute the Final Temperature

Add \( -36 \) and \( 68 \) to find the temperature. \( T(10) = 32 \). Therefore, the temperature at a height of \( 10,000 \) feet is \( 32 ^{\circ} \mathrm{F} \).

Key Concepts

Temperature Conversion

Temperature conversion is not only about switching between units like Celsius and Fahrenheit, it also involves understanding how temperature varies with different factors, such as altitude. In the given exercise, the formula describes how temperature changes with height above the Earth's surface in Fahrenheit degrees.
Understanding the temperature formula, \( T(h) = -3.6h + 68 \), can help us predict the temperature at different altitudes. Here, \( T(h) \) represents the temperature at a height \( h \) in thousands of feet.
Since the reduction factor is \(-3.6\), it shows how much the temperature drops for each increment of height. This formula is linear, as the change in temperature is consistent for changes in altitude.
To make accurate predictions, ensure you're using the correct units, like thousands of feet in this formula. This concept emphasizes how crucial it is to grasp units and factors affecting temperature when converting or calculating weather conditions.

Evaluation of Functions

In mathematics, evaluating a function means finding its output for a given input. For our exercise, we had to evaluate the function \( T(h) = -3.6h + 68 \) to find the temperature at a specific height.
The steps to evaluate this function include:

• Converting the given information to the correct units. For context, \( 10,000 \) feet was converted to \( 10 \) thousand feet, suitable for the formula.
• Substituting the numerical value of the height into the function. Here, we replaced \( h \) with \( 10 \) in the function.
• Performing arithmetic operations to simplify the expression. For instance, multiplying and then adding numbers in the formula: \(-3.6 \times 10 + 68 \).
• Finalizing the calculation to get the answer. The result was \( 32^{\circ} \mathrm{F} \).

By evaluating functions, we can determine specific values for practical applications like temperature, ensuring that each step follows logically from the input to the result. This skill is essential for analyzing scenarios modeled by functions.

Mathematical Modeling

Mathematical modeling involves using math to represent, analyze, and solve real-world problems. In this case, the function \( T(h) = -3.6h + 68 \) models how temperature changes with altitude above the Earth's surface. It translates a physical phenomenon—temperature variation with height—into a mathematical expression.
The components of the model include:

• Understanding the situation, such as how temperature decreases with increasing altitude due to thinner air and less heat retention.
• Representing this situation with a mathematical equation that can predict temperature for any given altitude.
• Analyzing the equation to interpret results, like knowing that the slope \(-3.6\) indicates the rate of temperature decrease per thousand feet.
• Using this model to make predictions or solve problems, like finding the temperature at a given height.

Modeling is a powerful way to bridge the gap between abstract math and concrete situations. It enables us to use equations to answer real-life questions and better understand the world around us, making complex phenomena manageable and comprehensible.