based on the half-life of Tc-99, how many half-lives have to pass for a 150 mg sample of Tc-99 to decay down to 30mg?

2.322 half-lives have passed to decay down from 150 miligrams to 30 miligrams.

Explanation:

The half of Technetium-99 is approximately 211000 years. The decay of isotopes is represented by the following ordinary differential equation:

(Eq. 1)

Where:

- First derivative of isotope mass in time, measured in miligrams per year.

- Mass of the isotope, measured in miligrams.

- Time constant, measured in years.

Now we proceed to obtain the solution of this differential equation:

(Eq. 2)

Where:

- Initial mass of the isotope, measured in miligrams.

- Time, measured in years.

The time passed for isotope is cleared within the equation described above:

In addition, we can obtain the time constant as a function of half-life:

(Eq. 3)

If we know that , and , then the time passed is:

The amount of passed half-lives is that time divided by a half-life. That is:

2.322 half-lives have passed to decay down from 150 miligrams to 30 miligrams.

in scientific notation, the value of the  speed of light in a vacuum

  will be written as 2.995 * 10^8. when this value is represented in the form of

  a * 10^b, then, a = 2.995 and b = 8.

the value of the speed of light in a vacuum given above can also be rounded up to approximately 300000000. in this case, the scientific notation will be written as 3 *10^8.