The Table Shows How An Elevator 500 Feet Above The Ground Is Descending At A Steady Rate.$[ \begin{tabular}{|c|c|} \hline \text{Time In Seconds} , (t) & \text{Height In Feet} , H(t) \ \hline 0 & 500 \ \hline 5 & 475 \ \hline 10 & 450
Understanding the Problem
The given table represents the height of an elevator above the ground at different time intervals. The table shows that the elevator is initially at a height of 500 feet and is descending at a steady rate. We are required to analyze the given data and determine the rate at which the elevator is descending.
Analyzing the Data
| Time in seconds (t) | Height in feet (h(t)) |
|---|---|
| 0 | 500 |
| 5 | 475 |
| 10 | 450 |
From the table, we can see that the elevator is descending at a steady rate. This means that the height of the elevator is decreasing at a constant rate. We can calculate the rate of descent by finding the difference in height over a given time interval.
Calculating the Rate of Descent
To calculate the rate of descent, we can use the formula:
Rate of descent = (Change in height) / (Change in time)
We can use the data from the table to calculate the rate of descent. Let's consider the time interval from 0 to 5 seconds.
Change in height = 500 - 475 = 25 feet Change in time = 5 - 0 = 5 seconds
Rate of descent = (25 feet) / (5 seconds) = 5 feet/second
Finding the Equation of the Height Function
Now that we have found the rate of descent, we can use it to find the equation of the height function. The height function is a function that describes the height of the elevator at any given time.
Let's assume that the height function is of the form:
h(t) = a + bt
where a and b are constants.
We know that the elevator is initially at a height of 500 feet, so we can write:
h(0) = 500 = a + b(0) a = 500
We also know that the rate of descent is 5 feet/second, so we can write:
h'(t) = 5
Differentiating the height function with respect to time, we get:
h'(t) = b
Equating this to the rate of descent, we get:
b = 5
Now that we have found the values of a and b, we can write the equation of the height function:
h(t) = 500 - 5t
Verifying the Equation
To verify the equation, we can use the data from the table to check if it satisfies the equation.
Let's consider the time interval from 0 to 5 seconds.
h(0) = 500 - 5(0) = 500 h(5) = 500 - 5(5) = 475
Both values match the data from the table, so we can conclude that the equation is correct.
Conclusion
In this article, we analyzed the given table and determined the rate at which the elevator is descending. We found that the rate of descent is 5 feet/second and used it to find the equation of the height function. We verified the equation by using the data from the table and found that it matches the given data.
Key Takeaways
- The table represents the height of an elevator above the ground at different time intervals.
- The elevator is descending at a steady rate.
- The rate of descent is 5 feet/second.
- The equation of the height function is h(t) = 500 - 5t.
Further Analysis
This problem can be further analyzed by considering different time intervals and calculating the rate of descent at each interval. We can also use the equation of the height function to find the height of the elevator at any given time.
Real-World Applications
This problem has real-world applications in the field of engineering and physics. For example, it can be used to design and optimize elevator systems in buildings. It can also be used to study the motion of objects in physics and engineering.
Mathematical Concepts
This problem involves the following mathematical concepts:
- Functions
- Derivatives
- Equations
- Graphs
Problem-Solving Strategies
This problem requires the following problem-solving strategies:
- Analyzing data
- Calculating rates
- Finding equations
- Verifying results
Conclusion
In conclusion, this problem requires the analysis of a table representing the height of an elevator above the ground at different time intervals. We determined the rate at which the elevator is descending and used it to find the equation of the height function. We verified the equation by using the data from the table and found that it matches the given data. This problem has real-world applications in the field of engineering and physics and involves the use of mathematical concepts such as functions, derivatives, and equations.
Q: What is the initial height of the elevator?
A: The initial height of the elevator is 500 feet.
Q: What is the rate of descent of the elevator?
A: The rate of descent of the elevator is 5 feet/second.
Q: How can we calculate the rate of descent?
A: We can calculate the rate of descent by finding the difference in height over a given time interval. For example, if we consider the time interval from 0 to 5 seconds, the change in height is 25 feet and the change in time is 5 seconds. Therefore, the rate of descent is (25 feet) / (5 seconds) = 5 feet/second.
Q: What is the equation of the height function?
A: The equation of the height function is h(t) = 500 - 5t, where h(t) is the height of the elevator at time t.
Q: How can we verify the equation of the height function?
A: We can verify the equation by using the data from the table to check if it satisfies the equation. For example, if we consider the time interval from 0 to 5 seconds, we can calculate the height of the elevator using the equation and compare it with the given data.
Q: What are some real-world applications of this problem?
A: This problem has real-world applications in the field of engineering and physics. For example, it can be used to design and optimize elevator systems in buildings. It can also be used to study the motion of objects in physics and engineering.
Q: What mathematical concepts are involved in this problem?
A: This problem involves the following mathematical concepts:
- Functions
- Derivatives
- Equations
- Graphs
Q: What problem-solving strategies are required to solve this problem?
A: This problem requires the following problem-solving strategies:
- Analyzing data
- Calculating rates
- Finding equations
- Verifying results
Q: Can we use this problem to design and optimize elevator systems?
A: Yes, this problem can be used to design and optimize elevator systems in buildings. By analyzing the rate of descent and the equation of the height function, we can design an elevator system that meets the required specifications.
Q: Can we use this problem to study the motion of objects in physics and engineering?
A: Yes, this problem can be used to study the motion of objects in physics and engineering. By analyzing the rate of descent and the equation of the height function, we can study the motion of objects and make predictions about their behavior.
Q: What are some potential limitations of this problem?
A: Some potential limitations of this problem include:
- Assuming a constant rate of descent
- Ignoring external factors such as air resistance
- Using a simplified model of the elevator system
Q: How can we address these limitations?
A: We can address these limitations by:
- Using more complex models that take into account external factors
- Incorporating more data and variables into the problem
- Using numerical methods to solve the problem
Q: What are some potential extensions of this problem?
A: Some potential extensions of this problem include:
- Adding more variables and data to the problem
- Using different mathematical models to describe the elevator system
- Incorporating real-world constraints and limitations into the problem
Q: How can we use this problem to teach mathematical concepts?
A: We can use this problem to teach mathematical concepts such as functions, derivatives, and equations. By analyzing the problem and solving it, students can develop a deeper understanding of these concepts and learn to apply them to real-world problems.
Q: What are some potential applications of this problem in education?
A: Some potential applications of this problem in education include:
- Using the problem to teach mathematical concepts in a real-world context
- Incorporating the problem into a curriculum or course
- Using the problem as a project or assignment for students.