Problem 70 - Brief Applied Calculus

Question

GENERAL: Boiling Point At higher altitudes, water boils at lower temperatures. This is why at high altitudes foods must be boiled for longer times - the lower boiling point imparts less heat to the food. At an altitude of \(h\) thousand feet above sea level, water boils at a temperature of \(B(h)=-1.8 h+212\) degrees Fahrenheit. Find the altitude at which water boils at \(98.6\) degrees Fahrenheit. (Your answer will show that at a high enough altitude, water boils at normal body temperature. This is why airplane cabins must be pressurized - at high enough altitudes one's blood would boil.)

Step-by-Step Solution

Answer

At 63,000 feet, water boils at 98.6°F.

Step 1: Understand the given formula

The problem gives a formula \( B(h) = -1.8h + 212 \) that shows how the boiling point of water varies with altitude \( h \), measured in thousands of feet. \( B(h) \) represents the boiling temperature.

Step 2: Setup the equation for the boiling point

We need to find the altitude \( h \) at which water boils at \( 98.6 \) degrees Fahrenheit. Therefore, set \( B(h) = 98.6 \). The equation becomes \( -1.8h + 212 = 98.6 \).

Step 3: Solve the equation for h

Start by subtracting \( 212 \) from both sides to isolate the term with \( h \):\[-1.8h = 98.6 - 212\]Simplify the right side:\[-1.8h = -113.4\].

Step 4: Isolate and solve for h

Divide both sides by \(-1.8\) to solve for \( h \):\[h = \frac{-113.4}{-1.8}\]Calculate the result:\[h = 63\].

Step 5: Interpret the result

The solution, \( h = 63 \), means at an altitude of 63 thousand feet, water boils at 98.6 degrees Fahrenheit.

Key Concepts

Boiling Temperature Formula

To understand how altitude affects boiling point, we use a specific formula. This formula helps find the boiling temperature of water at different heights above sea level. At sea level, water boils at 212 degrees Fahrenheit. However, the boiling point decreases as altitude increases. The formula given in the exercise is:

• \( B(h) = -1.8h + 212 \)

Here,

• \( B(h) \) represents the boiling temperature at altitude \( h \).
• \( h \) is the altitude, measured in thousands of feet.

The negative sign indicates that as \( h \) increases, \( B(h) \) decreases. Essentially, for every thousand-foot increase in elevation, the boiling temperature decreases by 1.8 degrees Fahrenheit. Understanding this formula is crucial for calculating how altitude affects boiling points.

Altitude Effects on Boiling

As you get higher in altitude, the atmospheric pressure decreases. This means there is less external pressure on the water's surface. With less pressure, water can boil at a lower temperature. The higher you climb, the less heat is required to make water boil. This is why boiling water at high altitudes happens at temperatures below the usual 212 degrees Fahrenheit.

• The formula \( B(h) = -1.8h + 212 \) shows this relationship. It predicts the boiling point at varying altitudes.
• For instance, at 63,000 feet, the boiling point drops to 98.6 degrees Fahrenheit, which is close to human body temperature.

At such high altitudes, your blood could boil without pressurization. This explains why airplane cabins must maintain a specific pressure. By knowing how altitude affects boiling, adjustments can be made in cooking and safety practices.

Solving Linear Equations

In the context of this exercise, solving linear equations is key to finding the boiling point of water at specific altitudes. A linear equation is one where the highest power of any variable is one, as depicted in our formula \( B(h) = -1.8h + 212 \).To solve for \( h \), the steps are straightforward:

• Set the equation equal to your target boiling point, such as 98.6 degrees Fahrenheit.
• Subtract 212 from both sides: \(-1.8h = 98.6 - 212\).
• Simplify to \(-1.8h = -113.4\).
• Divide by \(-1.8\) to find \( h \).

Thus, \( h = 63 \). This means at 63,000 feet, water boils at about body temperature. Understanding how to rearrange and solve these types of equations is a fundamental skill in math. It has practical applications, such as determining boiling points in this context.