the amount of radioactive element remaining, r, in a 100 - mg sample after d days is represented using the…

Explanation:

Step1: Recall the general form of exponential - decay

The general form of exponential decay is $y = a(1 - r)^t$, where $a$ is the initial amount, $r$ is the rate of decay per - time period, and $t$ is the number of time periods. We are given $r = 100(\frac{1}{2})^{\frac{d}{5}}$. Let's rewrite it in the general form for a one - day time period. When $d = 1$, $r_1=100(\frac{1}{2})^{\frac{1}{5}}$.

Step2: Calculate the value of $(\frac{1}{2})^{\frac{1}{5}}$

$(\frac{1}{2})^{\frac{1}{5}}=2^{-\frac{1}{5}}$. Using a calculator, $2^{-\frac{1}{5}}\approx0.8706$.

Step3: Find the decay rate

If $y = a(1 - r)^t$, and when $t = 1$, $y=a(1 - r)$. Here, $a = 100$ and $y = 100\times0.8706$ (from step 2). So, $1-r = 0.8706$. Then $r=1 - 0.8706=0.1294$.

Answer:

$12.94%$