The amount of plutonium given in a sample is determined by the function:
A(t)=2.00e^-0.053
t= years How do I figure the rate of decay after four years?
To figure out the rate of decay after four years, you need to take the derivative of the given function A(t) with respect to time (t). The derivative will give you the rate of change of the amount of plutonium over time.
Given function: A(t) = 2.00e^(-0.053t)
To find the derivative, you can use the chain rule. The chain rule states that if you have a composite function, f(g(x)), the derivative of that function is the derivative of the outer function multiplied by the derivative of the inner function.
Let's differentiate each part of the function step by step:
1. Differentiate the exponential function: d/dx(e^x) = e^x
In this case, we have e^(-0.053t). So, by applying the chain rule:
d/dt(e^(-0.053t)) = e^(-0.053t) * (-0.053) = -0.053e^(-0.053t)
2. Multiply by the coefficient: 2.00 * -0.053e^(-0.053t) = -0.106e^(-0.053t)
Therefore, the rate of decay after four years, denoted as A'(t), is -0.106e^(-0.053t).
To find the specific rate of decay after four years, substitute t = 4 into the derived function:
A'(4) = -0.106e^(-0.053 * 4)