Question

Researchers are monitoring two different radioactive substances. They have 300 grams of substance A which decays at a rate of 0.15%. They have 500 grams of substance B which decays at a rate of 0.37%. They are trying to determine how many years it will be before the substances have an equal mass.
If M represents the mass of the substance and t represents the elapsed time in years, then which of the following systems of equations can be used to determine how long it will be before the substances have an equal mass?

Answer 1

Answer:

The system of equations is:

$M_A(t)=300(0.9985)^t\\ \\ M_B(t)=500(0.9963)^t$

Explanation:

A. Substance A.

For substance A the rate of decay is 0.15% per year meaning that every year the mass is multiplied by a factor of 1 -0.15/100 = 1 - 0.0015.

Thus, as they have initially 300 g of substance A, its mass can be modeled by the function:

• $M_A(t)=300(1-0.0015)^t=300(0.9985)^t$

Where    $M_A(t)$    is the mass as a function of time, and t is the number of years elapsed.

B. Substance B.

For substance B the rate of decay is 0.37% per year meaning that every year the mass is multiplied by a factor of 0.37/100 = 0.0037.

As they have initially 500 g of substance B, its mass can be modeled by the function:

• $M_B(t)=500(1-0.0037)^t=500(0.9963)^t$

Where    $M_B(t)$    is the mass as a function of time, and t is the number of years elapsed.

C. System of equations:

$M_A(t)=300(0.9985)^t\\ \\ M_B(t)=500(0.9963)^t$

If you make    $M_A(t)=M_B(t)\\ \\ 300(0.9985)^t=500(0.9963)^t$    

You can solve for t, the time when the sustances have equal mass.