Write a formula relating y to t.
Use exact expressions to fill in the missing parts of the formula.
Do not use approximations. in the form as y = ___ e ^(

The formula relating \( y \) (the amount of substance remaining at time \( t \)) to \( t \) (time elapsed) is derived from the exponential decay model. Given that the half-life of the substance is 4 hours, we can express the decay constant \( k \) exactly as \( \frac{\ln(2)}{4} \).

Thus, the formula is:

\[
y = 89.8 \cdot e^{\left(-\frac{\ln(2)}{4} \cdot t\right)}
\]

Here:
- \( 89.8 \) is the initial amount of the substance.
- \( \frac{\ln(2)}{4} \) is the exact decay constant.
- \( t \) is the time in hours.

So the complete formula is:

\[
y = 89.8 \cdot e^{\left(-\frac{\ln(2)}{4} \cdot t\right)}
\]

Would you like any further clarification or details? Here are some related questions you might find interesting:

1. How can we use this formula to find the time it takes for the substance to decay to half its original amount?
2. What is the significance of the natural logarithm \( \ln(2) \) in the context of half-life?
3. How would the formula change if the initial amount were different?
4. How can this formula be used to predict the amount of substance left after any given time?
5. What if the substance had a different decay constant? How would that affect the formula?
6. Can this formula be applied to other types of decay, like biological decay?
7. What happens to the formula as \( t \) approaches infinity?
8. How could we graph this equation to visualize the decay process?

**Tip:** Remember that the exact expression for the decay constant is crucial for maintaining accuracy in your calculations, especially when dealing with exponential decay over long periods.

Understand the exact formula for exponential decay relating y to t, where y is the substance remaining and t is time elapsed. Learn how to derive and apply this formula accurately with a half-life of 4 hours. Includes detailed explanations and examples.

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