Creating a hypothetical formula for a type of plastic that decays by 1% per year can involve the use of decay models. One common way to represent decay mathematically is through an exponential decay formula.
Exponential Decay Formula
The general formula for exponential decay is:
\[ N(t) = N_0 \times e^{-kt} \]
Where:
- \( N(t) \) is the amount of material remaining at time \( t \).
- \( N_0 \) is the initial quantity of the material.
- \( k \) is the decay constant.
- \( t \) is the time in years.
- \( e \) is the base of the natural logarithm (approximately equal to 2.71828).
For 1% Decay Per Year
If we specify that the material decays by 1% per year, the relationship can be described as follows:
1% decay means that after 1 year, 99% of the material remains.
Thus, we can set up the equation for 1 year:
\[ N(1) = 0.99 \times N_0 \]
Using the exponential decay formula:
\[ N_0 \times e^{-k \cdot 1} = 0.99 \times N_0 \]
Dividing both sides by \( N_0 \):
\[ e^{-k} = 0.99 \]
To solve for \( k \), you can take the natural logarithm of both sides:
\[ -k = \ln(0.99) \]
So:
\[ k = -\ln(0.99) \]
Calculating \( k \):
\[ k \approx -(-0.01005) \approx 0.01005 \]
The Final Formula
Now, substituting \( k \) back into the exponential decay formula, the formula for the plastic that decays by 1% per year would be:
\[ N(t) = N_0 \times e^{-0.01005t} \]
This formula allows you to calculate the remaining amount of the plastic over time \( t \) in years, with an annual decay rate of 1%.
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