Question

In the half-life function Q(t)=28550⋅((34)h)(th) Q ( t ) = 28550 ⋅ ( ( 3 4 ) h ) ( t h ) , what is the half-life, h h , if (34)h=12 ( 3 4 ) h = 1 2 ?

Answer 1

Answer:

$h=2.4 hours \: 2hours\:24'$

Step-by-step explanation:

1)Rewriting it properly:

$Q(t)=28850*\left ( \left ( \frac{3}{4}\right )^{h} \right )^{\frac{t}{h}}\, if \left ( \frac{3}{4} \right )^{h}=\frac{1}{2}$

2) Let's calculate the time (in hours), based on this relation:

$\left ( \frac{3}{4} \right )^{h}=\frac{1}{2} \Rightarrow log_{\frac{3}{4}}\frac{1}{2} \Rightarrow h \approx 2.4\: hours$

3) Testing it. We must find something around the half of 28850, due to some rounding in logarithms.

$Q(t)=28850*\left ( \left ( \frac{3}{4}\right )^{h} \right )^{\frac{t}{h}}\, if \left ( \frac{3}{4} \right )^{h}=\frac{1}{2}\Rightarrow h=\\Q(t)=28850(\frac{1}{2})^{\frac{t}{h}}\Rightarrow Q(t)=28850(\frac{1}{2})^{\frac{t}{2.4}}\\28850(\frac{1}{2})^{\frac{t}{2.4}}=14425 \Rightarrow (\frac{1}{2})^{\frac{t}{2.4}}=\frac{14425}{28850}\Rightarrow (\frac{1}{2})^{\frac{t}{2.4}}=\frac{1}{2}\Rightarrow t=2.4\\Q(2.4)=28850*\left ( \left ( \frac{3}{4}\right )^{2.4} \right )^{\frac{2.4}{2.4}}\Rightarrow Q\approx14464$

4) So, h≈ 2.40 hours or 2 hours 24'