Question

the half - life of a radioactive element is five years. a scientist has 18 grams of the element. the equation representing the number of grams, g, after x years is $g = 18(0.5)^{\frac{x}{5}}$. what is the annual rate of decay? 3% 13% 87% 97%

Explanation:

Step1: Recall the general decay formula

The general formula for exponential decay is $g = g_0(1 - r)^x$, where $g_0$ is the initial amount, $r$ is the rate of decay per - time unit, and $x$ is the number of time units. In our case, the formula is $g = 18(0.5)^{\frac{x}{5}}$. Let's consider the case when $x = 1$.

Step2: Set up the equation for one - year decay

When $x = 1$, we have $g_1=18(0.5)^{\frac{1}{5}}$. And from the general formula $g_1 = 18(1 - r)^1$. So, $(1 - r)=(0.5)^{\frac{1}{5}}$.

Step3: Calculate $(0.5)^{\frac{1}{5}}$

$(0.5)^{\frac{1}{5}}=\sqrt[5]{0.5}\approx0.8706$.

Step4: Solve for $r$

If $1 - r = 0.8706$, then $r=1 - 0.8706 = 0.1294\approx 13\%$.

Answer:

13%