A Bouncing Ball Reaches A Height Of 54 Inches At Its First Peak, 36 Inches At Its Second Peak, And 24 Inches At Its Third Peak. Which Formula Represents This Scenario?A. $f(x) = 54\left(\frac{2}{3}\right)^x$B. $f(x) =
Introduction
When a ball bounces, it follows a predictable pattern of heights, with each peak being lower than the previous one. This phenomenon can be modeled using mathematical functions, specifically exponential decay. In this article, we will explore the formula that represents the bouncing ball scenario, where the ball reaches a height of 54 inches at its first peak, 36 inches at its second peak, and 24 inches at its third peak.
Understanding Exponential Decay
Exponential decay is a mathematical concept that describes how a quantity decreases over time. In the context of the bouncing ball, the height of the ball decreases exponentially with each bounce. The formula for exponential decay is given by:
f(x) = a * (r)^x
where:
- a is the initial value (the height of the first peak)
- r is the decay rate (a constant that determines how quickly the height decreases)
- x is the number of bounces (or the time period)
Analyzing the Given Data
We are given the following data:
- First peak: 54 inches
- Second peak: 36 inches
- Third peak: 24 inches
We can see that the height of the ball decreases by a factor of 2/3 from one peak to the next. This suggests that the decay rate r is 2/3.
Finding the Formula
Using the given data and the formula for exponential decay, we can find the formula that represents this scenario. We know that the initial value a is 54 inches (the height of the first peak). We also know that the decay rate r is 2/3.
Substituting these values into the formula, we get:
f(x) = 54 * (2/3)^x
This formula represents the bouncing ball scenario, where the ball reaches a height of 54 inches at its first peak, 36 inches at its second peak, and 24 inches at its third peak.
Comparison with Other Options
Let's compare our formula with the other options given:
A. f(x) = 54 * (2/3)^x
This formula matches our derived formula, and it correctly represents the bouncing ball scenario.
B. f(x) = 54 * (3/2)^x
This formula is incorrect, as it represents an exponential growth scenario rather than a decay scenario.
Conclusion
In conclusion, the formula that represents the bouncing ball scenario is f(x) = 54 * (2/3)^x. This formula accurately models the exponential decay of the ball's height with each bounce, and it matches the given data.
Frequently Asked Questions
Q: What is the initial value (a) in the formula?
A: The initial value a is 54 inches, which is the height of the first peak.
Q: What is the decay rate (r) in the formula?
A: The decay rate r is 2/3, which determines how quickly the height decreases with each bounce.
Q: How does the formula represent the bouncing ball scenario?
A: The formula represents the bouncing ball scenario by modeling the exponential decay of the ball's height with each bounce, using the given data.
Final Thoughts
The bouncing ball scenario is a classic example of exponential decay, and it can be modeled using mathematical functions. By understanding the formula that represents this scenario, we can gain insights into the behavior of the ball and make predictions about its future behavior.
Introduction
In our previous article, we explored the formula that represents the bouncing ball scenario, where the ball reaches a height of 54 inches at its first peak, 36 inches at its second peak, and 24 inches at its third peak. We derived the formula f(x) = 54 * (2/3)^x and compared it with other options. In this article, we will answer some frequently asked questions about the bouncing ball scenario and provide additional insights.
Q&A
Q: What is the significance of the decay rate (r) in the formula?
A: The decay rate r determines how quickly the height of the ball decreases with each bounce. In this scenario, the decay rate is 2/3, which means that the height of the ball decreases by a factor of 2/3 with each bounce.
Q: How does the formula account for the initial value (a)?
A: The initial value a is 54 inches, which is the height of the first peak. The formula f(x) = 54 * (2/3)^x takes into account the initial value and models the exponential decay of the ball's height with each bounce.
Q: Can the formula be used to predict the height of the ball at future bounces?
A: Yes, the formula can be used to predict the height of the ball at future bounces. By plugging in the value of x, which represents the number of bounces, we can calculate the height of the ball at any given bounce.
Q: What is the relationship between the decay rate (r) and the height of the ball?
A: The decay rate r is inversely proportional to the height of the ball. As the decay rate increases, the height of the ball decreases, and vice versa.
Q: Can the formula be used to model other types of exponential decay?
A: Yes, the formula f(x) = a * (r)^x can be used to model other types of exponential decay, such as population growth or chemical reactions.
Q: What is the significance of the exponent (x) in the formula?
A: The exponent x represents the number of bounces or the time period. As x increases, the height of the ball decreases exponentially.
Q: Can the formula be used to model the bouncing ball scenario with different initial values?
A: Yes, the formula f(x) = a * (r)^x can be used to model the bouncing ball scenario with different initial values. Simply plug in the new initial value a and the decay rate r to get the new formula.
Additional Insights
The Role of Exponential Decay
Exponential decay is a fundamental concept in mathematics and physics. It describes how a quantity decreases over time, and it is used to model a wide range of phenomena, including population growth, chemical reactions, and financial markets.
The Importance of Initial Values
The initial value a is a critical component of the formula f(x) = a * (r)^x. It determines the starting point of the exponential decay, and it has a significant impact on the behavior of the function.
The Relationship between Decay Rate and Height
The decay rate r is inversely proportional to the height of the ball. As the decay rate increases, the height of the ball decreases, and vice versa. This relationship is a fundamental aspect of exponential decay and is used to model a wide range of phenomena.
Conclusion
In conclusion, the formula f(x) = 54 * (2/3)^x represents the bouncing ball scenario, where the ball reaches a height of 54 inches at its first peak, 36 inches at its second peak, and 24 inches at its third peak. We have answered some frequently asked questions about the formula and provided additional insights into the behavior of the ball.