Question

1. a radioactive material has a mass given by $m(t)=126(0.84)^t$, where the mass is in grams and the time, $t$, is in years. which of the following gives the average rate of change of the mass over the interval $2\leq t \leq 5$ years?

(1) $-12.1$ grams per year
(2) $-36.2$ grams per year
(3) $18.3$ grams per year
(4) $28.9$ grams per year

Explanation:

Step1: Define average rate formula

The average rate of change of a function $m(t)$ over $[a,b]$ is $\frac{m(b)-m(a)}{b-a}$.
Here, $a=2$, $b=5$, $m(t)=126(0.84)^t$.

Step2: Calculate $m(5)$

Compute the mass at $t=5$:
$m(5)=126(0.84)^5$
$0.84^5 \approx 0.84\times0.84\times0.84\times0.84\times0.84 \approx 0.418$
$m(5) \approx 126\times0.418 \approx 52.67$

Step3: Calculate $m(2)$

Compute the mass at $t=2$:
$m(2)=126(0.84)^2$
$0.84^2=0.7056$
$m(2)=126\times0.7056 \approx 88.91$

Step4: Compute average rate

Substitute values into the formula:
$\frac{m(5)-m(2)}{5-2}=\frac{52.67-88.91}{3}=\frac{-36.24}{3}\approx -12.1$

Answer:

(1) -12.1 grams per year