Question

An element with a mass of 740 grams decays by 28.4% per minute. To the nearest minute, how long will it be until there are 30 grams of the element remaining?

Answer 1

Answer:t=9.6 minutes

Step-by-step explanation:

Given

mass of element is $A_o=740\ gm$

Decay rate $r=28.4\%\ \text{per minute}$

and we know

Amount of exponential decay is given by

$A=A_o[1-r]^t$

Where $A_0=\text{Initial amount}$

$\text{A=Accumulated amount}$

$\text{r=rate}$

$\text{t=time}$

Substituting values we get

$\Rightarrow 30=740[1-0.284]^t$

$\Rightarrow \dfrac{30}{740}=0.716^t$

$\Rightarrow 0.0405=0.716^t$

taking log both sides we get

$\Rightarrow\ \ln (0.0405)=t\times \ln (0.716)$

$\Rightarrow t=\dfrac{\ln (0.0405)}{\ln (0.716)}$

$\Rightarrow t=9.598\approx 9.6\ \text{minutes}$