Question

a scientist recorded the amount of substance z present in an ongoing chemical reaction. this table gives the amount of substance z over the first 15 minutes. the data can be modeled using an exponential function.

time (min)	3	6	9	12	15

based on the data, which measurement is closest to the amount of substance z that will be present after 30 minutes?

• 0 g
• 0.08 g
• 0.14 g
• 0.46 g

Explanation:

Step1: Define decay function

The general exponential decay model is $A(t) = A_0 \cdot r^{\frac{t}{k}}$, where $A_0$ is initial amount, $r$ is decay factor per $k$ minutes, $t$ is time.

Step2: Find decay factor per 3 min

Use $t=3$, $A(3)=6.2$; $t=6$, $A(6)=3.84$. Calculate $r = \frac{3.84}{6.2} \approx 0.619$.

Step3: Find initial amount $A_0$

Rearrange formula: $A_0 = \frac{A(3)}{r} = \frac{6.2}{0.619} \approx 10.02$ g.

Step4: Calculate amount at 30 min

Substitute $t=30$, $k=3$, $A_0\approx10.02$, $r\approx0.619$:
$A(30) = 10.02 \cdot (0.619)^{\frac{30}{3}} = 10.02 \cdot (0.619)^{10}$
Calculate $(0.619)^{10} \approx 0.008$, so $A(30) \approx 10.02 \cdot 0.008 \approx 0.08$ g.

Answer:

0.08 g