Incense Stick Burn Time Calculation

Problem Conundrum

The burn time of an incense stick depends on the following factors:

• The composition and ratio of combustible ingredients, which affect the burn rate.
• The environmental conditions, particularly the presence of wind or breeze, which can accelerate the burning process.

Given these factors, we aim to calculate the duration it takes for an incense stick to completely burn out under different ingredient ratios and environmental conditions. The main challenges are:

• Quantifying the effect of each ingredient on burn time.
• Predicting the impact of altered ratios on overall burn time.
• Estimating the decrease in burn time in the presence of a slight breeze.

Mathematical Modeling and Methodology

1. Burn Rate Formula

We define the burn rate \( R \) (in units of length per time) based on the composition of ingredients. If the incense stick has a length \( L \) and burns out in time \( T \), then: \[ R = \frac{L}{T} \]

2. Effect of Ingredient Ratios

Let the incense stick be composed of ingredients \( A \), \( B \), and \( C \), with ratios \( a \), \( b \), and \( c \) respectively. The burn rate \( R \) can be approximated as a function of these ratios: \[ R = k (a \cdot A + b \cdot B + c \cdot C) \] where \( k \) is a proportionality constant based on the material properties.

3. Calculating Burn Time for Different Ratios

By varying the values of \( a \), \( b \), and \( c \), we can calculate the corresponding burn times: \[ T = \frac{L}{R} = \frac{L}{k (a \cdot A + b \cdot B + c \cdot C)} \]

4. Impact of Breeze

Introducing a slight breeze increases the burn rate by a factor \( \delta \) (e.g., 10% increase in rate), where: \[ R_{\text{breeze}} = R \times (1 + \delta) \] Thus, the new burn time under breeze conditions is: \[ T_{\text{breeze}} = \frac{L}{R_{\text{breeze}}} = \frac{L}{R \times (1 + \delta)} \]

Conclusion

By applying the above equations, we can calculate the burn duration for an incense stick based on different ingredient compositions and environmental conditions. This approach provides insight into how ingredient ratios and external factors like wind impact the overall burn time.