Question

the radioactive element radium decays at a rate of 2.4% annually. if there are 50 grams of radium present, how much radium remains after 20 years (to the nearest tenth)?

Explanation:

Step1: Identify the decay formula

The formula for exponential decay is $A = P(1 - r)^t$, where $P$ is the initial amount, $r$ is the rate of decay, and $t$ is the time. Here, $P = 50$ grams, $r=0.024$, and $t = 20$.

Step2: Substitute the values into the formula

$A=50\times(1 - 0.024)^{20}=50\times(0.976)^{20}$.

Step3: Calculate $(0.976)^{20}$

Using a calculator, $(0.976)^{20}\approx0.61878$.

Step4: Calculate the remaining amount

$A = 50\times0.61878 = 30.939\approx30.9$ grams.

Answer:

$30.9$ grams