Problem 40 - Algebra and Trigonometry

Question

Heat of a Campfire 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 he is 20 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 approximately 25.2 ft from the fire.

Step 1: Understand the relationship

The heat experienced by the hiker is proportional to the amount of wood on the fire and inversely proportional to the cube of the distance from the fire. This relationship can be expressed as:\[H = rac{W}{d^3}\]where \(H\) is the heat felt, \(W\) is the amount of wood, and \(d\) is the distance from the fire.

Step 2: Initial condition setup

Initially, the hiker is 20 ft from the fire. We denote the original amount of wood as \(W_1\) and the original distance as \(d_1 = 20\) ft. The heat experienced is given by:\[H_0 = \frac{W_1}{20^3}\]

Step 3: New condition with doubled wood

When the wood is doubled, the new amount of wood \(W_2 = 2W_1\). To find the new distance \(d_2\), we use the equation:\[H = \frac{2W_1}{d_2^3}\]We want this to equal the original heat, \(H_0\).

Step 4: Setting the equations equal

Set the new heat equation equal to the old heat equation to find the relationship between the distances:\[\frac{2W_1}{d_2^3} = \frac{W_1}{20^3}\]Dividing both sides by \(W_1\), we get:\[\frac{2}{d_2^3} = \frac{1}{20^3}\]

Step 5: Solve for the new distance

To solve for \(d_2\), rearrange the equation:\[d_2^3 = 2 \times 20^3\]Now compute \(d_2\):\[d_2 = \sqrt[3]{2 \times 20^3}\]This simplifies to:\[d_2 = \sqrt[3]{16000}\]Calculating gives approximately:\[d_2 \approx 25.2 \, \text{ft}\]

Step 6: Verify the solution

Verify by checking that \(d_2\) satisfies the condition:- The original heat was \(H_0 = \frac{W_1}{20^3}\).- The heat at \(d_2 = 25.2\) ft with doubled wood is \(\frac{2W_1}{(25.2)^3}\).This ensures the heat remains the same, verifying our solution is correct.

Key Concepts

Proportionality

Understanding the concept of proportionality is crucial in solving real-world problems like the heat from a campfire question. Proportionality refers to a relationship where one quantity increases or decreases as another does. In simpler terms, when the relationship between two quantities is proportional, if you multiply one quantity by a number, the other is multiplied by the same number.
In the context of our problem, the heat felt by the hiker is directly proportional to the amount of wood on the fire. This means that if you increase the wood, the heat increases, and if you decrease the wood, the heat decreases. The equation to express this is: \[ H \propto W \]where \( H \) is heat and \( W \) is the wood. Increasing the wood amount will double the heat, under the assumption distance remains constant.
On the flip side, the heat from the campfire is inversely proportional to the cube of the distance. This means as the distance increases, the heat decreases, following the relationship:\[ H \propto \frac{1}{d^3} \]
Understanding how these types of proportionality work helps us manage and manipulate the conditions to find the equilibrium, such as maintaining the same heat when the wood amount changes.

Cubic Relationship

A cubic relationship is essentially a relationship where a variable is proportional to the cube of another variable. In our exercise, the heat experienced by the hiker is related to the cube of the distance from the campfire. This is denoted in the problem as: \[ H = \frac{W}{d^3} \]Here, \(d^3\) represents the cube of the distance, which is a significant factor in determining the heat's strength.
The cube of a number means multiplying the number by itself twice. For instance, \( d^3 = d \times d \times d \). This implies that as distance increases, its impact increases dramatically because the increase is cubed. In our campfire exercise, a small increase in distance can have a substantial decrease in heat experienced, due to the cubic relationship.
In practical terms:

• If the distance is doubled, the heat reduces by a factor of \( 2^3 = 8 \).
• If the original distance is tripled, the heat reduces to \( \frac{1}{27} \) of the original heat.

Recognizing this cubic relationship allows us to solve for distance changes when wood amount changes, as it plays a critical role in balancing heat perception.

Distance and Heat Relationship

The relationship between distance and heat, particularly in the context of a source like a campfire, can be understood through the laws of physics associated with heat dispersion and distance. When you are closer to the source—the campfire in this instance—you feel more heat. This connection describes how spread-out energy diminishes its intensity over a distance.
The equation governing this exercise is:\[ H = \frac{W}{d^3} \]With this, the heat \( H \) decreases as the distance \( d \) increases due to the inverse proportionality to the cube of \( d \).
This inverse cubic relationship means:

• The farther away you move, the exponentially less heat you feel, because \( d^3 \) rapidly increases with distance.
• To maintain the same heat when the wood amount doubles, the hiker must move further away, to a distance where the increased wood compensates for the changing cube effect of distance.

By correctly adjusting distance, we can ensure the heat felt remains constant despite changes in wood or initial factors. Understanding these principles is key in various practical applications, like adjusting seating distances from heat sources to achieve desired warmth.