Problem 70 - 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 thousand feet.

Step 1: Identify the Boiling Point Function

We are given the function for the boiling point of water at varying altitudes: \( B(h) = -1.8h + 212 \). The function represents the boiling temperature of water \( B(h) \) in degrees Fahrenheit at an altitude of \( h \) thousand feet above sea level.

Step 2: Set Up the Equation to Solve for Altitude

We need to find the altitude \( h \) at which water boils at 98.6 degrees Fahrenheit. To do this, we set the boiling point function equal to 98.6: \( -1.8h + 212 = 98.6 \).

Step 3: Rearrange to Isolate the Altitude Term

Subtract 212 from both sides to isolate the term with \( h \): \( -1.8h = 98.6 - 212 \). Simplify the right-hand side to find \( -1.8h = -113.4 \).

Step 4: Solve for Altitude, h

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

Step 5: Interpret the Result

The altitude at which water boils at 98.6 degrees Fahrenheit is 63 thousand feet above sea level.

Key Concepts

Boiling Point Function

The boiling point function is a mathematical model representing how water's boiling temperature changes based on altitude. At sea level, water boils at 212 degrees Fahrenheit. However, as altitude increases, atmospheric pressure decreases, and water boils at lower temperatures.

In our exercise, we are given the boiling point function as \( B(h) = -1.8h + 212 \). This equation tells us the boiling temperature \( B(h) \) of water, in degrees Fahrenheit, at an altitude of \( h \) thousand feet above sea level.

The slope of \(-1.8\) indicates that with every 1,000 feet increase in altitude, the boiling point decreases by 1.8 degrees Fahrenheit. This linear relationship is crucial in understanding how altitude affects water's boiling point. It supports the idea that at higher altitudes, food must boil for longer periods due to the lower boiling temperatures.

Solving Linear Equations

Solving linear equations is a fundamental skill in algebra where the goal is to find the value of a variable that makes the equation true. In this problem, we start with the equation \(-1.8h + 212 = 98.6\). This equation arises when we want to find the altitude at which water boils at 98.6 degrees Fahrenheit.

The first step is to rearrange the equation to isolate the variable \(h\). We subtract 212 from both sides, resulting in \(-1.8h = 98.6 - 212\). Simplifying further gives \(-1.8h = -113.4\).

Next, we solve for \(h\) by dividing both sides by \(-1.8\). This yields \(h = \frac{-113.4}{-1.8}\), and upon calculation, \(h\) equals 63. Thus, water would boil at 98.6 degrees Fahrenheit at an altitude of 63 thousand feet above sea level.

This exercise demonstrates the use of basic algebraic manipulation to derive a meaningful real-world conclusion from a mathematical expression.

Temperature Variation with Altitude

Temperature variation with altitude refers to the change in temperature as one moves higher above sea level. This concept is important in contexts like cooking at high altitudes and understanding the environmental conditions encountered during high-altitude flights.

When we examine this concept through the exercise, the boiling point of water decreases as altitude increases. This is because at higher altitudes, the atmospheric pressure is lower, so water molecules require less energy to escape into a vapor state, thus boiling occurs at a lower temperature.

Practically, this means that at really high altitudes like 63 thousand feet, water can boil at the temperature of the human body's normal range, such as 98.6 degrees Fahrenheit. This is why measures like pressurizing airplane cabins are essential to prevent bodily fluids from boiling at such heights. Overall, understanding how temperature varies with altitude helps in planning better for situations like mountain expeditions or aviation operations.