The Table Shows The Height Of A Candle As It Is Continuously Burned.Candle Height$[ \begin{tabular}{|c|c|} \hline \text{Time (hours)} & \text{Height (cm)} \ \hline 0 & 25 \ \hline 0.25 & 24.375 \ \hline 0.5 & 23.75 \ \hline 0.75 & 23.125
Understanding the Problem
The given table represents the height of a candle at different time intervals as it is continuously burned. The table provides us with the time in hours and the corresponding height of the candle in centimeters. Our task is to analyze this data and understand the pattern or relationship between the time and the height of the candle.
Analyzing the Data
Let's start by examining the data provided in the table. We can see that the initial height of the candle is 25 cm at time 0 hours. As time progresses, the height of the candle decreases. At 0.25 hours, the height is 24.375 cm, at 0.5 hours it is 23.75 cm, and at 0.75 hours it is 23.125 cm.
Calculating the Rate of Change
To understand the rate at which the height of the candle is decreasing, we can calculate the difference in height between consecutive time intervals.
- Between 0 and 0.25 hours, the height decreases by 0.625 cm.
- Between 0.25 and 0.5 hours, the height decreases by 0.625 cm.
- Between 0.5 and 0.75 hours, the height decreases by 0.625 cm.
Identifying the Pattern
From the calculations above, we can see that the height of the candle is decreasing at a constant rate of 0.625 cm per 0.25 hours. This means that the height of the candle is decreasing by 2.5 cm per hour.
Modeling the Situation
To model the situation, we can use a linear equation to represent the relationship between the time and the height of the candle. Let's assume that the height of the candle at time t is given by the equation:
h(t) = 25 - 2.5t
where h(t) is the height of the candle at time t and t is the time in hours.
Verifying the Model
To verify the model, we can substitute the given values of time and height into the equation and check if it holds true.
- At time 0 hours, the height is 25 cm. Substituting t = 0 into the equation, we get h(0) = 25 - 2.5(0) = 25, which is true.
- At time 0.25 hours, the height is 24.375 cm. Substituting t = 0.25 into the equation, we get h(0.25) = 25 - 2.5(0.25) = 24.375, which is true.
- At time 0.5 hours, the height is 23.75 cm. Substituting t = 0.5 into the equation, we get h(0.5) = 25 - 2.5(0.5) = 23.75, which is true.
- At time 0.75 hours, the height is 23.125 cm. Substituting t = 0.75 into the equation, we get h(0.75) = 25 - 2.5(0.75) = 23.125, which is true.
Conclusion
In conclusion, the table shows that the height of the candle is decreasing at a constant rate of 2.5 cm per hour. We can model this situation using a linear equation, which accurately represents the relationship between the time and the height of the candle.
Mathematical Representation
The mathematical representation of the situation can be given by the equation:
h(t) = 25 - 2.5t
where h(t) is the height of the candle at time t and t is the time in hours.
Graphical Representation
The graphical representation of the situation can be given by a straight line with a negative slope, where the x-axis represents the time and the y-axis represents the height of the candle.
Real-World Applications
The concept of a candle burning at a constant rate has many real-world applications, such as:
- Calculating the time it takes for a candle to burn completely.
- Determining the height of a candle at a given time.
- Modeling the behavior of other objects that burn at a constant rate.
Limitations
The model assumes that the candle burns at a constant rate, which may not be true in reality. Other factors such as air pressure, temperature, and humidity may affect the rate at which the candle burns.
Future Work
Future work can involve:
- Investigating the effect of air pressure, temperature, and humidity on the rate at which the candle burns.
- Developing a more accurate model that takes into account these factors.
- Applying the concept of a candle burning at a constant rate to other real-world problems.
Conclusion
In conclusion, the table shows that the height of the candle is decreasing at a constant rate of 2.5 cm per hour. We can model this situation using a linear equation, which accurately represents the relationship between the time and the height of the candle. The concept of a candle burning at a constant rate has many real-world applications, and future work can involve investigating the effect of other factors on the rate at which the candle burns.
Q: What is the initial height of the candle?
A: The initial height of the candle is 25 cm.
Q: At what rate is the height of the candle decreasing?
A: The height of the candle is decreasing at a constant rate of 2.5 cm per hour.
Q: How can we model the situation using a linear equation?
A: We can model the situation using the equation h(t) = 25 - 2.5t, where h(t) is the height of the candle at time t and t is the time in hours.
Q: What is the height of the candle at time 0.25 hours?
A: The height of the candle at time 0.25 hours is 24.375 cm.
Q: What is the height of the candle at time 0.5 hours?
A: The height of the candle at time 0.5 hours is 23.75 cm.
Q: What is the height of the candle at time 0.75 hours?
A: The height of the candle at time 0.75 hours is 23.125 cm.
Q: Can we use this model to predict the height of the candle at any given time?
A: Yes, we can use this model to predict the height of the candle at any given time by substituting the value of t into the equation h(t) = 25 - 2.5t.
Q: What are some real-world applications of this concept?
A: Some real-world applications of this concept include calculating the time it takes for a candle to burn completely, determining the height of a candle at a given time, and modeling the behavior of other objects that burn at a constant rate.
Q: What are some limitations of this model?
A: Some limitations of this model include the assumption that the candle burns at a constant rate, which may not be true in reality. Other factors such as air pressure, temperature, and humidity may affect the rate at which the candle burns.
Q: Can we develop a more accurate model that takes into account these factors?
A: Yes, we can develop a more accurate model that takes into account these factors by incorporating them into the equation h(t) = 25 - 2.5t.
Q: What are some future directions for this research?
A: Some future directions for this research include investigating the effect of air pressure, temperature, and humidity on the rate at which the candle burns, developing a more accurate model that takes into account these factors, and applying the concept of a candle burning at a constant rate to other real-world problems.
Q: Can we use this concept to solve other problems in mathematics and science?
A: Yes, we can use this concept to solve other problems in mathematics and science, such as modeling the behavior of other objects that burn at a constant rate, calculating the time it takes for a candle to burn completely, and determining the height of a candle at a given time.
Q: What are some potential applications of this concept in engineering and technology?
A: Some potential applications of this concept in engineering and technology include designing and optimizing candle-burning systems, developing more efficient candle-burning technologies, and applying the concept of a candle burning at a constant rate to other engineering and technological problems.
Q: Can we use this concept to solve problems in other fields, such as economics and finance?
A: Yes, we can use this concept to solve problems in other fields, such as economics and finance, by applying the concept of a candle burning at a constant rate to other economic and financial problems, such as modeling the behavior of stock prices or calculating the time it takes for a financial instrument to mature.