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? y = 25(0.15)^t y = 25(0.85)^t y = 50(0.15)^t y = 50(0.85)^t

Explanation:

Step1: Recall decay - function 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: Analyze half - life information

The half - life is 4 hours. We can also use the formula $y=a\cdot b^{\frac{t}{h}}$, where $h$ is the half - life. Another way is to find the decay factor for 1 hour. If the half - life is 4 hours, we know that when $t = 4$, $y=\frac{a}{2}$. Let's assume the function is $y = a\cdot b^t$. When $a = 50$ and $t = 4$, $y = 25$. So, $25=50\cdot b^4$. Then $b^4=\frac{25}{50}=0.5$. Solving for $b$, we get $b = 0.5^{\frac{1}{4}}\approx0.84$.

Step3: Select the correct function

The closest value to our calculated $b$ value among the options is when $b = 0.85$ and $a = 50$. So the function is $y = 50(0.85)^t$.

Answer:

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