the half - life of a certain substance is about 4 hours. the graph shows the decay of a 50 gram sample of…

Explanation:

Step1: Recall decay - formula

The general formula for exponential decay is $y = a(1 - r)^t$, where $a$ is the initial amount, $r$ is the rate of decay, and $t$ is the time. The initial amount of the substance is $a = 50$ grams.

Step2: Calculate the decay - rate using half - life

The half - life is 4 hours. After 4 hours, the amount of the substance is half of the initial amount. Let's use the formula $y=a(1 - r)^t$. When $t = 4$ and $y=\frac{a}{2}$, and $a = 50$, we have $\frac{50}{2}=50(1 - r)^4$. Divide both sides by 50: $\frac{1}{2}=(1 - r)^4$. Take the fourth - root of both sides: $1 - r=\sqrt[4]{\frac{1}{2}}\approx0.84$. So, $r\approx1 - 0.84 = 0.16\approx0.15$.

Step3: Write the decay function

Substitute $a = 50$ and $r\approx0.15$ into the decay formula $y=a(1 - r)^t$. We get $y = 50(0.85)^t$.

Answer:

$y = 50(0.85)^t$