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 starts at a height of 500 feet and descends at a steady rate. We are asked to analyze the situation and find the rate at which the elevator is descending.
The Table
| Time in seconds (t) | Height in feet h(t) |
|---|---|
| 0 | 500 |
| 5 | 475 |
| 10 | 450 |
Analyzing the Situation
To find the rate at which the elevator is descending, we need to find the rate of change of the height with respect to time. This can be represented mathematically as the derivative of the height function h(t) with respect to time t.
The Height Function
Let's assume that the height function h(t) is a linear function, which means that the rate of change of the height is constant. This is a reasonable assumption, given that the elevator is descending at a steady rate.
Finding the Rate of Change
To find the rate of change of the height, we need to find the derivative of the height function h(t) with respect to time t. We can do this by using the definition of a derivative:
h'(t) = lim(h(t + Δt) - h(t)) / Δt
where Δt is a small change in time.
Using the Table
We can use the table to find the rate of change of the height. Let's consider the time interval from 0 to 5 seconds. During this time, the height of the elevator changes from 500 feet to 475 feet.
Calculating the Rate of Change
To calculate the rate of change of the height, we can use the formula:
h'(t) = (h(t + Δt) - h(t)) / Δt
where Δt = 5 seconds.
h'(0) = (h(5) - h(0)) / 5 = (475 - 500) / 5 = -25 / 5 = -5
Conclusion
The rate of change of the height of the elevator is -5 feet per second. This means that the elevator is descending at a rate of 5 feet per second.
Generalizing the Result
We can generalize the result by finding the rate of change of the height at any time t. We can do this by using the definition of a derivative:
h'(t) = lim(h(t + Δt) - h(t)) / Δt
where Δt is a small change in time.
Using the Table
We can use the table to find the rate of change of the height at any time t. Let's consider the time interval from t to t + Δt. During this time, the height of the elevator changes from h(t) to h(t + Δt).
Calculating the Rate of Change
To calculate the rate of change of the height, we can use the formula:
h'(t) = (h(t + Δt) - h(t)) / Δt
where Δt is a small change in time.
h'(t) = (h(t + Δt) - h(t)) / Δt
Simplifying the Expression
We can simplify the expression by using the fact that the height function h(t) is linear.
h(t + Δt) = h(t) + h'(t) * Δt
Substituting this expression into the formula for h'(t), we get:
h'(t) = (h(t) + h'(t) * Δt - h(t)) / Δt
Simplifying the expression, we get:
h'(t) = h'(t)
This means that the rate of change of the height is constant, and we can use the value of h'(0) to find the rate of change of the height at any time t.
Conclusion
The rate of change of the height of the elevator is -5 feet per second. This means that the elevator is descending at a rate of 5 feet per second.
Final Answer
The final answer is feet per second.
Frequently Asked Questions
We have received many questions about the table that shows how an elevator 500 feet above the ground is descending at a steady rate. Here are some of the most frequently asked questions and our answers:
Q: What is the initial height of the elevator?
A: The initial height of the elevator is 500 feet.
Q: What is the height of the elevator after 5 seconds?
A: The height of the elevator after 5 seconds is 475 feet.
Q: What is the height of the elevator after 10 seconds?
A: The height of the elevator after 10 seconds is 450 feet.
Q: What is the rate of change of the height of the elevator?
A: The rate of change of the height of the elevator is -5 feet per second.
Q: Why is the rate of change of the height of the elevator constant?
A: The rate of change of the height of the elevator is constant because the elevator is descending at a steady rate.
Q: Can the rate of change of the height of the elevator be different at different times?
A: No, the rate of change of the height of the elevator is constant and does not change at different times.
Q: How can we find the rate of change of the height of the elevator at any time t?
A: We can find the rate of change of the height of the elevator at any time t by using the formula:
h'(t) = (h(t + Δt) - h(t)) / Δt
where Δt is a small change in time.
Q: What is the significance of the rate of change of the height of the elevator?
A: The rate of change of the height of the elevator is significant because it tells us how fast the elevator is descending.
Q: Can the rate of change of the height of the elevator be positive or negative?
A: The rate of change of the height of the elevator is negative because the elevator is descending.
Q: What is the final height of the elevator?
A: The final height of the elevator is not given in the table, but we can find it by using the formula:
h(t) = h(0) + h'(t) * t
where t is the time in seconds.
Conclusion
We hope that this Q&A article has helped to clarify any questions you may have had about the table that shows how an elevator 500 feet above the ground is descending at a steady rate. If you have any further questions, please don't hesitate to ask.
Final Answer
The final answer is feet per second.