A substance decays so that the amount a of the substance left at time t is given by: a = a0 ∙ (0.8)t where a0 is the original amount of the substance. what is the half-life (the amount of time that it takes to decay to half the original amount) of this substance rounded to the nearest tenth of a year?

The exponential decline, which is a rapid reduction over time, can be calculated with the use of the exponential decay formula.
The exponential decay formula is used to determine population decay, half-life, radioactivity decay, and other phenomena.
The general form is F(x) = a.

Here,

a = the initial amount of substance

1-r is the decay rate

x = time span

The equation is given in its correct form as follows:

a = [tex]a_{0}[/tex]×[tex](0.8)^{t}[/tex]

As this is an exponential decay of a first order reaction, t is an exponent of 0.8.

Now let's figure out the half life. Since the amount left is half of the initial amount at time t, that is when:

a = 0.5 a0

0.5[tex]a_{0}[/tex] = [tex]a_{0}[/tex]×[tex](0.8)^{t}[/tex]

0.5 = [tex](0.8)^{t}[/tex]

taking log on both sides

t log 0.8 = log 0.5

t = log 0.5/log 0.8

t = 3.106 years

The half-life of the substance is 3.106 years.

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