To determine the constant [tex]\(a\)[/tex] in the equation [tex]\(A(t) = A_0 \cdot a^t\)[/tex], we need to connect this with the known half-life of chromium-49, which is 42.3 minutes.
The half-life of a substance is the time it takes for half of the substance to decay. This can be represented with the formula:
[tex]\[
A(t) = A_0 \cdot \left(\frac{1}{2}\right)^{\frac{t}{\text{half-life}}}
\][/tex]
Here, the half-life is 42.3 minutes. Therefore, the formula becomes:
[tex]\[
A(t) = A_0 \cdot \left(\frac{1}{2}\right)^{\frac{t}{42.3}}
\][/tex]
We need to express this in the form [tex]\(A(t) = A_0 \cdot a^t\)[/tex]. To do this, we need to find [tex]\(a\)[/tex] such that:
[tex]\[
a^t = \left(\frac{1}{2}\right)^{\frac{t}{42.3}}
\][/tex]
To isolate [tex]\(a\)[/tex], consider [tex]\(t = 1\)[/tex] minute:
[tex]\[
a = \left(\frac{1}{2}\right)^{\frac{1}{42.3}}
\][/tex]
Calculating this value:
[tex]\[
a = 2^{-\frac{1}{42.3}}
\][/tex]
The value of [tex]\(2^{-\frac{1}{42.3}}\)[/tex] rounded to six decimal places is approximately:
[tex]\[
a \approx 0.983747
\][/tex]
Thus, the value of [tex]\(a\)[/tex] that describes the amount of chromium-49 left after [tex]\(t\)[/tex] minutes, rounded to six decimal places, is:
[tex]\[
\boxed{0.983747}
\][/tex]
