# Once a certain medicine is in the bloodstream, its half-life is 21 hours. How long will it be before an initial 30cc of the medicine has been reduced to 7cc?

Nov 18, 2016

Here is the formula given by my teacher to solve this problem.

$y = A {b}^{\frac{t}{k}}$

#### Explanation:

A represents the initial amount, in this case, it is 30cc

B is the base, half-life is always 1/2

And t represents time, we are told that it half every 21 hours, hence it should be $\frac{t}{21}$

$7 \mathcal{=} 30 \mathcal{\cdot} {\left(\frac{1}{2}\right)}^{\frac{t}{21}}$

$\left(\frac{7}{30}\right) = {\left(\frac{1}{2}\right)}^{\frac{t}{21}}$

$\log \left(\frac{7}{30}\right) = \left(\frac{t}{21}\right) \log \left(\frac{1}{2}\right)$

$\log \frac{\frac{7}{30}}{\log} \left(\frac{1}{2}\right) = \left(\frac{t}{21}\right)$

$t = 44.09$

The answer is round to two decimal places.