A particular compound decays according to the equation `y=ae^(-0.097t)`, where t is in days. Find the half-life of this compound.
Half-life is the amount of time it takes for the compound to decay to the half the amount there was originally.
Originally, or at the moment with t = 0, the amount of the compound was
`y=ae^(-0.97*0) = a`
So, half of that is `y=a/2` .
To find time t when this happens, plug this value of y into the equation:
`a/2 = ae^(-0.97t)`
Divide both sides by a:
`1/2 = e^(-0.97t)`
`ln(1/2) = -0.97t`
`t=(ln(1/2))/(-0.97) = (ln(2))/0.97`
`t = 0.71 `
The half-life of this compound is approximately 0.71 days.
Half life: `t_h=1/2 y(0)=a/2`
`t_h= ln 2/0.097` `=7^d 3^h 30^m 01^s 20^c`
Let you see tree line of compund decays
Red curve line: a=50, black curve line: a=30, blue curve line : a=20
Relative colored straight linea stand for half life decays:
(Red straight line : y= 25, Black straight line: y=15, Blue straight line: y=10)
Note, as calculus shown, the intersection of half life with rispective curve of decays line, does'nt depend by a, indeed the point of intersection have the same value of `t_h`