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Answer :
The function [tex]\( f(t) = 360(0.9935)^t \)[/tex] describes how a quantity changes over time, specifically over [tex]\( t \)[/tex] months. Here’s a step-by-step explanation of what the constant 0.9935 in this equation tells us about the rate of change:
1. Identify the Equation Type:
The function is in the form [tex]\( f(t) = a(b)^t \)[/tex], where:
- [tex]\( a = 360 \)[/tex] is the initial quantity.
- [tex]\( b = 0.9935 \)[/tex] is the base or growth/decay factor.
- [tex]\( t \)[/tex] represents the time in months.
2. Understanding the Growth/Decay Factor:
- The value [tex]\( b = 0.9935 \)[/tex] represents how the quantity changes each month.
- If [tex]\( b \)[/tex] is less than 1, this indicates a decay, meaning the quantity decreases over time.
- If [tex]\( b \)[/tex] was greater than 1, it would indicate growth.
3. Calculate the Rate of Decay:
- The rate of decay is determined by finding out how much [tex]\( b \)[/tex] is less than 1.
- This is done using the formula: [tex]\( \text{Decay rate} = 1 - b \)[/tex].
4. Find the Decay Rate:
- Substituting the value of [tex]\( b = 0.9935 \)[/tex] into the formula, we get:
[tex]\[ \text{Decay rate} = 1 - 0.9935 = 0.0065 \][/tex]
5. Interpret the Result:
- The decay rate of 0.0065 means that the quantity decreases by approximately 0.65% each month.
- This is a small but consistent decrease over each period.
In summary, the constant 0.9935 reveals that the quantity is undergoing a decay, and the rate of decay is about 0.65% per month.
1. Identify the Equation Type:
The function is in the form [tex]\( f(t) = a(b)^t \)[/tex], where:
- [tex]\( a = 360 \)[/tex] is the initial quantity.
- [tex]\( b = 0.9935 \)[/tex] is the base or growth/decay factor.
- [tex]\( t \)[/tex] represents the time in months.
2. Understanding the Growth/Decay Factor:
- The value [tex]\( b = 0.9935 \)[/tex] represents how the quantity changes each month.
- If [tex]\( b \)[/tex] is less than 1, this indicates a decay, meaning the quantity decreases over time.
- If [tex]\( b \)[/tex] was greater than 1, it would indicate growth.
3. Calculate the Rate of Decay:
- The rate of decay is determined by finding out how much [tex]\( b \)[/tex] is less than 1.
- This is done using the formula: [tex]\( \text{Decay rate} = 1 - b \)[/tex].
4. Find the Decay Rate:
- Substituting the value of [tex]\( b = 0.9935 \)[/tex] into the formula, we get:
[tex]\[ \text{Decay rate} = 1 - 0.9935 = 0.0065 \][/tex]
5. Interpret the Result:
- The decay rate of 0.0065 means that the quantity decreases by approximately 0.65% each month.
- This is a small but consistent decrease over each period.
In summary, the constant 0.9935 reveals that the quantity is undergoing a decay, and the rate of decay is about 0.65% per month.
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