Problem 46 - Precalculus Mathematics for Calculus

Question

The heat experienced by a hiker at a campfire is proportional to the amount of wood on the fire and inversely proportional to the cube of his distance from the fire. If the hiker is \(20 \mathrm{ft}\) from the fire and someone doubles the amount of wood burning, how far from the fire would he have to be so that he feels the same heat as before?

Step-by-Step Solution

Answer

The hiker should be 25 ft from the fire.

Step 1: Understanding the Relationship

The heat experienced, let's call it \( H \), is proportional to the amount of wood \( W \) and inversely proportional to the cube of the distance \( d \) from the fire. This can be expressed as \( H \propto \frac{W}{d^3} \), or more specifically, \( H = k \frac{W}{d^3} \) where \( k \) is a constant of proportionality.

Step 2: Initial Situation Model

Initially, the hiker experiences a certain heat at 20 ft with a certain amount of wood. Let's represent this initial condition as \( H_1 = k \frac{W}{20^3} \).

Step 3: After Doubling the Wood

The amount of wood is doubled, so the new heat \( H_2 \) is \( H_2 = k \frac{2W}{d_2^3} \). We want \( H_1 = H_2 \).

Step 4: Set Equations Equal

Set the expressions for \( H_1 \) and \( H_2 \) equal to each other to find \( d_2 \): \( k \frac{W}{20^3} = k \frac{2W}{d_2^3} \).

Step 5: Solve for New Distance

Cancel the constant \( k \) and the wood \( W \) from each side: \( \frac{1}{20^3} = \frac{2}{d_2^3} \). Solve for \( d_2^3 \): \( d_2^3 = 2 imes 20^3 \). Therefore, \( d_2 = \sqrt[3]{2 imes 20^3} \).

Step 6: Calculate the Value

Calculate \( 20^3 = 8000 \). Then \( d_2^3 = 16000 \), thus \( d_2 = \sqrt[3]{16000} \), which is 25.

Key Concepts

Inverse Proportionality

When we talk about inverse proportionality, we're dealing with two quantities that change in opposite directions. This means if one quantity increases, the other decreases, and vice versa. A simple example to understand this is through the equation: \( y \propto \frac{1}{x} \), which can also be written as \( y = \frac{k}{x} \). Here, \( k \) is a constant. As \( x \) grows larger, \( y \) becomes smaller because they are inversely related.
In the context of the hiker and the campfire, the heat (\( H \)) the hiker feels is inversely proportional to the cube of the distance (\( d^3 \)) he's standing from the fire. Simply put, as the hiker moves away, the heat reduces because \( d^3 \) increases, making \( H \) decrease if all other factors stay constant.

• The concept helps define how variables relate when one diminishes as the other rises.
• It models behaviors in natural phenomena, like how heat diminishes with distance.

Understanding inverse proportionality is crucial because it affects how we predict outcomes in real-world scenarios, including safe distances from heat sources.

Cube Relationship

The cube relationship concept is key when the rate of change affects three dimensions simultaneously. In mathematical terms, a cube relationship refers to a variable being proportional to the cube of another variable. For example, \( y \propto x^3 \) can be expressed as \( y = kx^3 \), where \( k \) is a constant.
In our campfire scenario, the cube of the distance (\( d^3 \)) impacts how heat is perceived by the hiker. The heat primarily depends on the ratio of the wood's amount and the distance cubed. The formula \( H = k \frac{W}{d^3} \) showcases how the cube affects heat inversely.

• This relationship implies a rapid change: shift a bit in distance, and the heat changes drastically.
• The effect is that moving away doubles the cube of the distance, noticeably reducing heat felt.

Cube relationships are important since they are part of the physics determining energy and distances, influencing how we interact with heat sources and maintain safety in temperature-effected spaces.

Distance and Heat Relationship

The distance and heat relationship illustrates how physical distance from a source affects heat intensity. As found in the problem statement, the heat (\( H \)) experienced is directly proportional to the amount of wood and inversely proportional to the cube of the distance (\( d^3 \)) from the fire. This relationship is crucial because it highlights two simultaneous dependencies:

• Direct Proportionality with Wood: More wood results in more heat, assuming distance is constant.
• Inverse Cubic Proportionality with Distance: Moving further away decreases the heat, as the distance exponentially reduces the heat intensity received.

This understanding is applied in our exercise when doubling the wood. The problem ensures the original heat felt remains the same even if wood quantity changes by adjusting the distance using the cube root concept. The way \( d_2 = \sqrt[3]{2 \times 20^3} \) balances out these variables by solving for the new distance \( d_2 \) needed to experience unchanged heat. Understanding this relationship is vital, as it informs how we manage spaces around fires for optimal safety and comfort.