The given function appears to be incorrectly formatted. The correct general form for exponential decay should be:<\/p>\n\n\n\n
[
P(t) = 100 \\cdot e^{-kt}
]<\/p>\n\n\n\n
or<\/p>\n\n\n\n
[
P(t) = 100 \\cdot a^t
]<\/p>\n\n\n\n
However, based on your notation, it seems like the decay model is written as:<\/p>\n\n\n\n
[
P(t) = 100(1.32)^t
]<\/p>\n\n\n\n
This function represents exponential growth rather than decay since the base is greater than 1. Typically, radioactive decay follows the form:<\/p>\n\n\n\n
[
P(t) = 100(a)^t, \\quad 0 < a < 1
]<\/p>\n\n\n\n
To determine the instantaneous rate of decay at the half-life, let’s proceed step by step:<\/p>\n\n\n\n
Step 1: Find the Half-Life<\/h3>\n\n\n\n
The half-life ( t_{1\/2} ) is the time when the substance is reduced to 50% of its original amount. That means:<\/p>\n\n\n\n
[
50 = 100(1.32)^t
]<\/p>\n\n\n\n
Solving for ( t ):<\/p>\n\n\n\n
[
\\frac{50}{100} = (1.32)^t
]<\/p>\n\n\n\n
[
0.5 = (1.32)^t
]<\/p>\n\n\n\n
Taking the natural logarithm on both sides:<\/p>\n\n\n\n
[
\\ln(0.5) = t \\ln(1.32)
]<\/p>\n\n\n\n
[
t = \\frac{\\ln(0.5)}{\\ln(1.32)}
]<\/p>\n\n\n\n
We will compute this value.<\/p>\n\n\n\n
Step 2: Compute Instantaneous Rate of Change<\/h3>\n\n\n\n
The derivative of ( P(t) ) is:<\/p>\n\n\n\n
[
P'(t) = 100 \\cdot \\ln(1.32) \\cdot (1.32)^t
]<\/p>\n\n\n\n
We substitute ( t_{1\/2} ) to get the instantaneous rate of decay at that moment.<\/p>\n\n\n\n
I will now calculate these values and generate an image.<\/p>\n\n\n\n The instantaneous rate of decay at the half-life is approximately -2.50<\/strong> (negative indicating decay), and the half-life of the substance is about 13.88 years<\/strong>. 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!<\/p>\n","protected":false},"excerpt":{"rendered":" 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. 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<\/figure>\n\n\n\n