Question

the half life of a certain substance is about 4 hours. the graph shows the decay of a 50 gram sample of the substance that is measured every hour for 9 hours. which function can be used to determine the approximate number of grams of the sample remaining after t hours? o y = 25(0.15)^t o y = 25(0.85)^t o y = 50(0.15)^t o y = 50(0.85)^t

Explanation:

Step1: Recall decay - function formula

The general form of an exponential - decay function 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 (since the sample starts with 50 grams).

Step2: Analyze the half - life information

The half - life is 4 hours. After 4 hours, the amount of the substance is approximately half of the initial amount. We can also use the fact that we can find the decay factor for 1 - hour intervals. If the half - life is 4 hours, we know that when $t = 4$, $y=\frac{a}{2}$. Let's find the decay factor $b$ in the formula $y = a\cdot b^t$. When $t = 4$ and $a = 50$, $y = 25$. So, $25=50\cdot b^4$. Then $b^4=\frac{25}{50}=0.5$. Solving for $b$, we get $b = 0.5^{\frac{1}{4}}\approx0.84$. Rounding to two decimal places, $b\approx0.85$.

Step3: Determine the function

Substituting $a = 50$ and $b\approx0.85$ into the exponential - decay function $y = a\cdot b^t$, we get $y = 50(0.85)^t$.

Answer:

$y = 50(0.85)^t$ (corresponding to the fourth option)