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

To model this situation we are going to use the exponential decay function: $f(t)=a(1-b)^t$
where
$f(t)$ is the final amount remaining after $t$ years of decay
$a$ is the initial amount
$b$ is the decay rate in decimal form 
$t$ is the time in years 

For substance A:
Since we have 300 grams of the substance, $a=300$. To convert the decay rate to decimal form, we are going to divide the rate by 100%:
$r= \frac{0.15}{100} =0.0015$. Replacing the values in our function:
$f(t)=a(1-b)^t$
$f(t)=300(1-0.0015)^t$
$f(t)=300(0.9985)^t$ equation (1)

For substance B:
Since we have 500 grams of the substance, $a=500$. To convert the decay rate to decimal form, we are going to divide the rate by 100%:
$r= \frac{0.37}{100} =0.0037$. Replacing the values in our function:
$f(t)=a(1-b)^t$
$f(t)=500(1-0.0037)^t$
$f(t)=500(0.9963)^t$ equation (2)

Since they are trying to determine how many years it will be before the substances have an equal mass$M$, we can replace $f(t)$ with $M$ in both equations:
$M=300(0.9985)^t$ equation (1)
$M=500(0.9963)^t$ equation (2) 

We can conclude that the system of equations that can be used to determine how long it will be before the substances have an equal mass, $M$, is:
$\left \{ {{M=300(0.9985)^t} \atop {M=500(0.9963)^t}} \right.$

Solving the system, we can show that it will take approximately 231.59 years for that to happen.

Answer 2

Answer: Guy who already answered this did it correctly, just forgot that it is decaying. Meaning the answer would be correct.. But the exponents need to be negative. I took this test on plato and got all of them correct.

Hope this could help a few people!