Finding The Value Of D For Exponential Decay
Exponential decay is a fundamental concept in mathematics and finds widespread applications in various scientific disciplines, including physics, chemistry, biology, and finance. At its core, exponential decay describes the phenomenon where a quantity decreases over time at a rate proportional to its current value. This means that the larger the quantity, the faster it decreases, and conversely, the smaller the quantity, the slower it decreases. In simpler terms, exponential decay represents a gradual decline in a quantity's magnitude as time progresses.
To fully grasp the concept of exponential decay, it's essential to understand the underlying mathematical principles. Exponential decay is mathematically modeled by an exponential function, which takes the form:
y(t) = a * e^(-kt)
where:
y(t)represents the quantity at timetais the initial quantity at timet = 0eis the base of the natural logarithm (approximately equal to 2.71828)kis the decay constant, a positive constant that determines the rate of decaytis time
The decay constant k plays a crucial role in determining the speed of the decay. A larger value of k indicates a faster rate of decay, while a smaller value of k corresponds to a slower rate of decay. The negative sign in the exponent ensures that the quantity decreases over time.
In the context of exponential decay, the domain represents the independent variable, typically time, while the range represents the dependent variable, which is the quantity that is decaying. The table you provided shows the relationship between the domain and the range for a specific exponential decay scenario. To determine the value of d that would make the table represent exponential decay, we need to analyze the pattern of decay and apply the principles of exponential functions.
Now, let's delve into the specific problem and determine the value of d that makes the table represent exponential decay.
In this section, we will delve into the heart of the problem: analyzing the given table to determine the value of d that ensures exponential decay. The table presents a set of domain-range pairs, and our goal is to identify the underlying pattern that governs the relationship between these pairs. To achieve this, we will carefully examine the changes in the range values as the domain values increase.
Understanding the Concept of Exponential Decay
Before we proceed, let's reiterate the essence of exponential decay. Exponential decay is characterized by a consistent proportional decrease in the dependent variable (range) for each unit increase in the independent variable (domain). In simpler terms, the quantity decreases by a fixed percentage over equal intervals of time or domain.
Examining the Given Data
The provided table presents the following domain-range pairs:
| Domain | Range |
|---|---|
| 0 | 32 |
| 1 | 24 |
| 2 | d |
From the table, we observe that when the domain increases from 0 to 1, the range decreases from 32 to 24. This decrease suggests a potential decay pattern. However, to confirm whether it's exponential decay, we need to verify if the decrease is proportional.
Calculating the Decay Factor
To determine the decay factor, we can divide the range value at domain 1 by the range value at domain 0:
Decay Factor = Range at Domain 1 / Range at Domain 0 = 24 / 32 = 3/4
This calculation reveals that the range decreases by a factor of 3/4 when the domain increases by 1. In exponential decay, this decay factor should remain constant for each unit increase in the domain.
Determining the Value of d
To find the value of d, we need to apply the same decay factor to the range value at domain 1. This means multiplying the range value at domain 1 (which is 24) by the decay factor (3/4):
d = Range at Domain 2 = Range at Domain 1 * Decay Factor = 24 * (3/4) = 18
Therefore, the value of d that makes the table represent exponential decay is 18.
Verifying Exponential Decay
To further confirm that the table represents exponential decay, we can check if the ratio between consecutive range values is constant:
- Ratio between Range at Domain 1 and Range at Domain 0: 24 / 32 = 3/4
- Ratio between Range at Domain 2 and Range at Domain 1: 18 / 24 = 3/4
The ratios are equal, which confirms that the table exhibits exponential decay.
Having established the principles of exponential decay and analyzed the given table, we now embark on the crucial step of calculating the precise value of d that ensures the table embodies exponential decay. This calculation involves applying the concept of a constant decay factor, which is a hallmark of exponential decay.
Recap of Exponential Decay
To reiterate, exponential decay signifies a consistent proportional decline in the range for each unit increment in the domain. This proportionality is quantified by the decay factor, which represents the ratio between successive range values. In essence, the range decreases by a fixed percentage for every unit increase in the domain.
Determining the Decay Factor
As we previously established, the decay factor can be computed by dividing the range value at a given domain by the range value at the preceding domain. In our table, we can calculate the decay factor using the range values at domains 0 and 1:
Decay Factor = Range at Domain 1 / Range at Domain 0 = 24 / 32 = 3/4
This calculation reveals that the range diminishes by a factor of 3/4 as the domain escalates by 1. In the realm of exponential decay, this decay factor remains constant across all unit increments in the domain.
Calculating the Value of d
To ascertain the value of d, we apply the identical decay factor to the range value at domain 1. This entails multiplying the range value at domain 1 (which is 24) by the decay factor (3/4):
d = Range at Domain 2 = Range at Domain 1 * Decay Factor = 24 * (3/4) = 18
Thus, the value of d that ensures the table represents exponential decay is 18.
Verification of Exponential Decay
To further validate that the table manifests exponential decay, we can verify the constancy of the ratio between consecutive range values:
- Ratio between Range at Domain 1 and Range at Domain 0: 24 / 32 = 3/4
- Ratio between Range at Domain 2 and Range at Domain 1: 18 / 24 = 3/4
These equal ratios affirm that the table exhibits exponential decay, thereby solidifying the accuracy of our calculated value for d.
Conclusion
By meticulously analyzing the table and applying the principles of exponential decay, we have successfully determined the value of d that ensures the table represents exponential decay. The calculated value of d is 18, which maintains the consistent proportional decrease in the range for each unit increase in the domain, a hallmark of exponential decay.
Therefore, the value of must be 18 for the table to represent exponential decay.
