Problem 61 - Algebra and Trigonometry with Analytic Geometry

Question

The temperature \(T\) (in \({ }^{\circ} \mathrm{C}\) ) at which water boils is related to the elevation \(h\) (in meters above sea level) by the formula $$ h=1000(100-T)+580(100-T)^{2} $$ for \(95 \leq T \leq 100\). (a) At what elevation does water boil at a temperature of \(98^{\circ} \mathrm{C}\) ? (b) The elevation of Mt. Everest is approximately 8840 meters. Estimate the temperature at which water boils at the top of this mountain. (Hint: Use the quadratic formula with \(x=100-T\).)

Step-by-Step Solution

Answer

(a) Elevation is 4320 meters. (b) Water boils at approximately 96.9°C.

Step 1: Substitute the temperature into the formula

For part (a), we need to determine the elevation \(h\) when the temperature \(T\) is \(98^{\circ} \mathrm{C}\). Substitute \(T = 98\) into the formula:\[h = 1000(100 - 98) + 580(100 - 98)^2\]Thus,\[h = 1000 \times 2 + 580 \times 4\]

Step 2: Calculate the elevation for part (a)

Now, calculate each term separately:- \(1000 \times 2 = 2000\)- \(580 \times 4 = 2320\)Add these together:\[h = 2000 + 2320 = 4320 \text{ meters}\]

Step 3: Set up the equation for part (b)

For part (b), we need to find the temperature \(T\) when the elevation \(h\) is 8840 meters. Set up the equation:\[8840 = 1000(100-T) + 580(100-T)^2\]Let \(x = 100 - T\). This gives:\[8840 = 1000x + 580x^2\]Rearrange to form a standard quadratic equation:\[580x^2 + 1000x - 8840 = 0\]

Step 4: Solve the quadratic equation

Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) with \(a = 580\), \(b = 1000\), and \(c = -8840\).Calculate the discriminant:\[b^2 - 4ac = 1000^2 - 4 \times 580 \times (-8840) = 1000000 + 20422400 = 21422400\]Then calculate \(x\):\[x = \frac{-1000 \pm \sqrt{21422400}}{2 \times 580}\]Calculate further:\[sqrt{21422400} \approx 4625.8\]\[x = \frac{-1000 \pm 4625.8}{1160}\]This gives two potential solutions; use the positive root:\[ x \approx \frac{3625.8}{1160} \approx 3.126 \]

Step 5: Determine the temperature at the mountain's elevation

Since we solved for \(x = 100 - T\), substitute back to find \(T\):\[100 - T = 3.126 \Rightarrow T = 100 - 3.126 = 96.874^{\circ} \mathrm{C}\]Thus, the boiling point of water on Mt. Everest is approximately \(96.9^{\circ} \mathrm{C}\).

Key Concepts

Quadratic Equations and Their Importance

Quadratic equations are mathematical expressions where the highest power of the unknown variable is two. Typically, such an equation is written in the form: \[ ax^2 + bx + c = 0 \] Here, 'a', 'b', and 'c' are constants, and 'x' represents the variable whose value is to be determined. Quadratic equations can appear in a range of real-world problems, including those involving projectile motion and optimization problems. They can be solved using methods such as factoring, completing the square, or the quadratic formula, which is extremely useful when the equation does not factor easily.

• The quadratic formula is given by: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
• The term \( \sqrt{b^2 - 4ac} \) is known as the discriminant, and it helps determine the number of solutions the equation has.

To solve the problem of finding the boiling point as mentioned in the original exercise, understanding and using quadratic equations is crucial. They help us find the value of the unknown variable, which in this context represents the relationship between temperature and elevation.

Temperature and Elevation Relationship

Temperature and elevation have an intriguing relationship where atmospheric pressure decreases as altitude increases. Water boils when the vapor pressure equals the atmospheric pressure. Therefore, at higher elevations like mountains, the boiling point of water reduces. For instance, if you boil water at sea level, it occurs around 100°C. However, at the elevation of Mt. Everest, atmospheric pressure is much lower, and therefore, water boils at a temperature significantly below 100°C. This phenomenon is crucial for cooking and scientific experiments at varying altitudes.

• As elevation increases, atmospheric pressure decreases.
• A lower atmospheric pressure results in a lower boiling point of water.

In the exercise, understanding how elevation affects boiling points is essential for solving both parts of the problem, giving practical insights into natural occurring temperature and pressure changes.

Understanding Boiling Point

The boiling point of a substance is the temperature at which it turns from a liquid to a gas. For water, this means the temperature at which water boils, changing from liquid to vapor. The boiling point can vary based on atmospheric pressure, which is why it differs with elevation.

• The standard boiling point of water is 100°C at 1 atmosphere of pressure (sea level).
• At higher elevations, such as on mountains, the boiling point is lower due to reduced pressure.

In the exercise, the boiling point is central to understanding at what temperature water will boil at different elevations. This concept allows us to predict and solve for different boiling temperatures encountered in high-altitude locations like Mt. Everest.

The Challenge of Mt. Everest

Mt. Everest represents one of the most extreme environments on Earth, standing at approximately 8,840 meters above sea level. Due to its altitude, the atmospheric pressure there is significantly lower than at sea level, affecting various environmental and human factors, including how water boils. This mountain's towering height means that boiling a pot of water at the summit can happen at below 100°C. Such unusual boiling temperatures can pose challenges in cooking and scientific operations, necessitating careful calculations to ensure successful outcomes. Understanding Mt. Everest's elevation provides context for finding the boiling point in such cases. The original exercise demonstrates practical application of quadratic equations to resolve boiling point questions pertinent to extreme elevation scenarios.