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

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

Explanation:

1)Rewriting it properly:

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

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

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

4) So, h≈ 2.40 hours or 2 hours 24'