Question

For the following in functions of the quantity of t minutes

Answer 1

Part A

Given

$Q(t)=400(2)^{\frac{t}{4}}$

Solution

We ask to double the time i.e

$\begin{gathered} 800=400(2)^{\frac{t}{4}} \\ \text{Divide both sides by 400} \\ \frac{800}{400}=\frac{400}{400}(2)^{\frac{t}{4}} \\ \\ 2^1=(2)^{\frac{t}{4}} \\ \\ 1=\frac{t}{4} \\ \\ \text{cross multiply} \\ t=1\times4 \\ t=4 \end{gathered}$

Part B

Given

$y=10(\frac{1}{2})^{\frac{t}{12}}$

Solution

The half-life would be 5

$\begin{gathered} y=10(\frac{1}{2})^{\frac{t}{12}} \\ 5=10(\frac{1}{2})^{\frac{t}{12}} \\ \text{divide both sides by 10} \\ \\ \frac{5}{10}=\frac{10}{10}(\frac{1}{2})^{\frac{t}{12}} \\ \\ \frac{1}{2}=(\frac{1}{2})^{\frac{t}{12}} \\ \\ 1=\frac{T}{12} \\ \text{cross multiply} \\ T=1\times12 \\ T=12 \end{gathered}$