Problem 82 - Algebra and Trigonometry

Question

Bonfire Temperature In the vicinity of a bonfire, the temperature \(T\) in \(^{\circ} \mathrm{C}\) at a distance of \(x\) meters from the center of the fire was given by $$ T=\frac{600,000}{x^{2}+300} $$ At what range of distances from the fire's center was the temperature less than \(500^{\circ} \mathrm{C} ?\)

Step-by-Step Solution

Answer

The temperature is less than 500°C when the distance is greater than 30 meters from the bonfire.

Step 1: Set up the inequality

To find the range of distances where the temperature is less than 500°C, begin by setting up the inequality \( \frac{600,000}{x^2 + 300} < 500 \).

Step 2: Clear the denominator

Multiply both sides of the inequality by \(x^2 + 300\) to eliminate the fraction: \(600,000 < 500(x^2 + 300)\).

Step 3: Expand and simplify

Distribute 500 on the right side of the inequality:\(600,000 < 500x^2 + 150,000\).

Step 4: Isolate the quadratic term

Subtract 150,000 from both sides to isolate the quadratic term:\(450,000 < 500x^2\).

Step 5: Solve for \(x^2\)

Divide both sides by 500:\(900 < x^2\).

Step 6: Find the solution for \(x\)

Take the square root of both sides to solve for \(x\):\( x > 30 \).

Key Concepts

Temperature Calculation

Temperature calculations can sometimes involve functions or equations to model how temperature changes with certain variables. In this scenario, the temperature near a bonfire is calculated using a specific mathematical model:

• The temperature in degrees Celsius, denoted as \(T\), depends on the distance \(x\) from the center of the bonfire.
• The relationship is given by the formula: \( T = \frac{600,000}{x^2 + 300} \).

This equation suggests that temperature changes as you move further from the fire.The quadratic term in the denominator indicates that the temperature decreases more gradually the farther you get. This inverse relationship helps us understand how heat dissipates over distance in real-life situations.To find a specific condition, like when the temperature is below a certain level (e.g., 500°C), we need to solve an inequality derived from the equation. We'll adjust the relationship to discover the range of distances fulfilling this condition.

Quadratic Equations

Quadratic equations frequently appear in mathematical modeling, showcasing relationships where variables are squared. In our problem, before solving for the distance, we're given a quadratic inequality:

• The inequality is initially framed as \( \frac{600,000}{x^2 + 300} < 500 \).
• By multiplying through to clear the fraction and expanding, it turns into a more common quadratic format.
• We end up with \( 500x^2 + 150,000 > 600,000 \), which simplifies to \( 500x^2 > 450,000 \).

Quadratics are essential in solving real-world problems, such as determining threshold conditions like distance or time. Solving this effectively means isolating \(x^2\) and taking the square root to yield \( x > 30 \). This step uncovers critical information like distance thresholds around phenomena like bonfires.

Distance Measurement

In physics and everyday situations, measurement of distance is crucial to many calculations and analyses. Here, we are examining how distance affects temperature perception near a bonfire.

• We started with the inequality \( \frac{600,000}{x^2 + 300} < 500 \), focusing on solving it for \(x\).
• Through calculations, it was found that \( x > 30 \) meters.
• This indicates that beyond 30 meters from the bonfire, the temperature falls below 500°C.

Distance isn't merely a static measure; it influences results in dynamic conditions, like diminishing heat feel due to dispersion. Understanding such effects aids in implications ranging from safe proximity around fires to planning spaces in safety measures, demonstrating how mathematical insights embody real-world applications.