Determine whether the statement is true or​ false, and explain why.
If a substance decays according to the equation y equals y 0 e Superscript kt Baseline comma
 
the constant k must be negative
.
Question content area bottom
Part 1
Choose the correct answer below.
A.
The statement is false
. 
If the equation is a decay​ equation, the value of k must be positive
.
B.
The statement is false. If the equation is a decay​ equation, the value of k can be any real number.
C.
The statement is true
. 
If the equation is a decay​ equation, the value of k must be negative.
D.
The statement is false. If the equation is a decay​ equation, the value of k must be zero.


The correct answer is **C. The statement is true. If the equation is a decay equation, the value of k must be negative.**

### Explanation:
The equation given is:

\[
y = y_0 e^{kt}
\]

- \(y_0\) is the initial amount of the substance.
- \(k\) is the rate constant.
- \(t\) is time.

For **decay**, we expect the quantity \(y\) to decrease over time. This happens when \(k\) is negative because:

\[
e^{kt} \quad \text{decreases as} \quad t \quad \text{increases, if} \quad k < 0.
\]

If \(k\) were positive, the exponential term would grow, implying **exponential growth**, not decay. Therefore, \(k\) must be negative for the substance to decay over time.

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### Do you need further details or have any other questions?

### Related Questions:
1. How does the value of \(k\) affect the rate of decay?
2. Can \(k\) ever be zero in a decay process?
3. What is the physical interpretation of \(y_0\) in the decay equation?
4. How would you solve for \(k\) if given data points for \(y\) and \(t\)?
5. How does exponential decay compare to linear decay in terms of speed?

### Tip:
In exponential decay, the quantity decreases by a fixed percentage over equal time intervals, unlike linear decay where it decreases by a fixed amount.

Determine whether the statement is true or false, and explain why. If a substance decays according to the equation y = y₀ e^(kt), the constant k must be negative.

Understand why the constant k must be negative in an exponential decay equation y = y₀ e^(kt). This solution explains the behavior of decay processes and explores the conditions for k in differential equations. Suitable for high school students learning exponential decay.

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