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a certain isotope decays so that the amount A remaining after t years is given by: A=A0 x e^-0.03t, where A0 is the original amount of the isotope. to the nearest year, the half-life of the isotope (the amount of the time it takes to decay to half the original amount) is ____ years

Respuesta :

carlosego

Answer:

[tex]t = 23.10\ years[/tex]

Step-by-step explanation:

We have an exponential decay function:

[tex]A = A_0e ^{-0.03t}[/tex]

This function decreases as the years pass. That is, if A is the quantity for a time t and [tex]A_0[/tex] is the original quantity at time t = 0 then:

[tex]A <A_0[/tex]

We want to know for what value of t the value of A decreases by half of  [tex]A_0[/tex]. That is, we want to know when [tex]A = \frac{A_0}{2}[/tex]

Then we do:

[tex]\frac{A_0}{2} = A_0e ^{-0.03t}\\\\\frac{1}{2} =e ^{-0.03t}\\\\ln(\frac{1}{2}) = -0.03t\\\\t = \frac{ln(\frac{1}{2})}{-0.03}[/tex]

[tex]t = 23.10\ years[/tex]

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