A certain radioactive substance decays exponentially. The percent, P, of the substance left after years is given by the function P() = 100(1.32). Determine the instantaneous rate of decay at the instant the half life of the substance is reached.
The Correct Answer and Explanation is :
The given function appears to be incorrectly formatted. The correct general form for exponential decay should be:
[
P(t) = 100 \cdot e^{-kt}
]
or
[
P(t) = 100 \cdot a^t
]
However, based on your notation, it seems like the decay model is written as:
[
P(t) = 100(1.32)^t
]
This function represents exponential growth rather than decay since the base is greater than 1. Typically, radioactive decay follows the form:
[
P(t) = 100(a)^t, \quad 0 < a < 1
]
To determine the instantaneous rate of decay at the half-life, let’s proceed step by step:
Step 1: Find the Half-Life
The half-life ( t_{1/2} ) is the time when the substance is reduced to 50% of its original amount. That means:
[
50 = 100(1.32)^t
]
Solving for ( t ):
[
\frac{50}{100} = (1.32)^t
]
[
0.5 = (1.32)^t
]
Taking the natural logarithm on both sides:
[
\ln(0.5) = t \ln(1.32)
]
[
t = \frac{\ln(0.5)}{\ln(1.32)}
]
We will compute this value.
Step 2: Compute Instantaneous Rate of Change
The derivative of ( P(t) ) is:
[
P'(t) = 100 \cdot \ln(1.32) \cdot (1.32)^t
]
We substitute ( t_{1/2} ) to get the instantaneous rate of decay at that moment.
I will now calculate these values and generate an image.
The instantaneous rate of decay at the half-life is approximately -2.50 (negative indicating decay), and the half-life of the substance is about 13.88 years. The graph above illustrates the decay function, highlighting the half-life point where the rate of change was computed. Let me know if you need any further clarification!