The mass of radioisotopes ( $m$ ), measured in milligrams, decreases exponentially in time ( $t$ ), measured in days. The model that represents such decrease is described below:

$m(t) = m_{o}\cdot e^{-\frac{t}{\tau} }$ (1)

Where:

$m_{o}$ - Initial mass, in milligrams.
$m(t)$ - Current mass, in milligrams.
$\tau$ - Time constant, in days.

In addition, the time constant is defined in terms of half-life ( $t_{1/2}$ ), in days:

$\tau = \frac{t_{1/2}}{\ln 2}$ (2)

If we know that $m_{o} = 85\,mg$ , $t_{1/2} = 27.7\,d$ and $m(t) = 8.5\,mg$ , then the time required for decaying is:

$\tau = \frac{t_{1/2}}{\ln 2}$

$\tau = \frac{27.7\,d}{\ln 2}$

$\tau \approx 39.963\,d$

$t = -\tau \cdot \ln \frac{m(t)}{m_{o}}$

$t = -(39.963\,d)\cdot \ln \frac{8.5\,mg}{85\,mg}$

$t\approx 92.018\,d$

About 92 days are taken for 90 % of the material to decay.

We kindly invite to check this question on half-life: brainly.com/question/24710827

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2 years ago