Problem 21 - 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 height if the temperature is \(-58^{\circ} \mathrm{F} .\)

Step-by-Step Solution

Answer

The height is 35 thousand feet.

Step 1: Understand the Problem

We need to find the height \( h \) at which the temperature \( T(h) \) is \(-58^{\circ} \mathrm{F}\). The function given is \( T(h) = -3.6h + 68 \).

Step 2: Set Up the Equation

Since we know the temperature \( T(h) \) is \(-58^{\circ} \mathrm{F}\), we can set the equation \( -3.6h + 68 = -58 \).

Step 3: Solve for \( h \)

We will solve the equation \( -3.6h + 68 = -58 \) for \( h \). First, subtract 68 from both sides to isolate the term with \( h \):\[ -3.6h = -58 - 68 \]This simplifies to:\[ -3.6h = -126 \].

Step 4: Simplify and Solve Division

Now, divide both sides by \(-3.6\) to find \( h \):\[ h = \frac{-126}{-3.6} \].

Step 5: Calculate the Result

Perform the division to find the value of \( h \):\[ h = 35 \].

Step 6: Interpret the Result

The height at which the temperature reaches \(-58^{\circ} \mathrm{F}\) is 35 thousand feet.

Key Concepts

Solving Equations

Solving equations is like finding the missing puzzle piece that makes everything fit together. When we solve an equation, we look for a value that can replace a variable, such as \( h \) in our case, that makes the equation true.
Understanding the steps to solve an equation is crucial because they help us think logically and process any equation systematically.

• Start by identifying what is known and what needs to be found. Here, we know the temperature, and we need to find the height.
• Set up the equation using the information given. In this example, we want \(-3.6h + 68 = -58\).
• Perform operations to isolate the variable. Subtract 68 from both sides to start solving \(-3.6h = -126\).
• Finally, divide to solve for the variable \( h = \frac{-126}{-3.6} \).

Apply these steps systematically to solve any linear equation with ease. Taking small steps transforms a complex problem into a manageable set of simple solutions.

Temperature Conversion

Temperature conversion is about transforming temperatures from one scale to another, like Celsius to Fahrenheit or vice versa. In this exercise, the function \( T(h) = -3.6h + 68 \) is about how temperature changes with height.
Unlike straightforward unit conversions, this function shows a linear relationship, meaning temperature changes steadily as height increases or decreases at a constant rate per thousand feet.

• The function tells us that for every 1,000 feet we go up, the temperature decreases by 3.6 degrees Fahrenheit.
• The starting temperature when the height \( h \) is zero is 68 degrees Fahrenheit.
• Solving the temperature function at a given height \(-58^{\circ} F\) also helps us find the height using the conversion logic embedded in the equation.

Instead of converting between measurement units, this exercise transitions our thoughts from a temperature scale to a temperature behavior based on altitude.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions to uncover the values embedded within them. This skill is pivotal in turning complex problems into simple ones, allowing us to isolate variables and solve for unknowns.
To solve for \( h \) in the function \( T(h) = -3.6h + 68 \), we need to perform a few algebraic steps:

• First, rearrange the equation \(-3.6h + 68 = -58\) to align terms for easy calculation.
• Subtract \( 68 \) from both sides to move the constant term away from the \( h \)-related terms, becoming \(-3.6h = -126\).
• Finally, divide both sides by \(-3.6\) to solve for \( h = 35 \).

Each step simplifies the equation, making it easier to solve, and highlights how careful manipulation of numbers and operations leads to clear solutions.