Respuesta :
Answer:
The expression that represents the number of days until only 10% remains is T((d) 10 %) =100×[tex](\frac{1}{2} )^{3.322}[/tex].
Step-by-step explanation:
The equation for half life is of the form
A = A₀×[tex](\frac{1}{2} )^{\frac{t}{h} }[/tex].........................................................................(1)
Where
A = Final amount
A₀ = Initial amount
t = Time
h = Half life
For the equation T(d) = 100×2⁽⁻²⁾....................................(2)
We have by comparison with the equation for half life
2 ≡ [tex]\frac{t}{h}[/tex] and and the equation (2) can be written as
Percentage remaining after 2 half lives is
[tex]\frac{A}{A_0}[/tex] ×100=100×[tex](\frac{1}{2} )^{2 }[/tex]
However if the half life of Technetium-99m is 6 hours then we have for one day
[tex]\frac{A}{A_0}[/tex] ×100=100×[tex](\frac{1}{2} )^{2 *2}[/tex]
Therefore an expression that represents the number of days until only 10% remains is
[tex]\frac{A}{A_0}[/tex] ×100=100×[tex](\frac{1}{2} )^{\frac{d}{h} }[/tex] = 10 %
[tex](\frac{1}{2} )^{\frac{d}{h} }[/tex] =[tex]\frac{1}{10}[/tex]
= ㏑[tex](\frac{1}{2} )^{\frac{d}{h} }[/tex] = ㏑([tex]\frac{1}{10}[/tex])
= [tex]\frac{d}{h}[/tex]×㏑[tex](\frac{1}{2} )[/tex] = ㏑([tex]\frac{1}{10}[/tex])
[tex]\frac{d}{h}[/tex] = [tex]\frac{ln(\frac{1}{10}) }{ln(\frac{1}{2} )}[/tex] = 3.322
Therefore the expression for the number of days 10 % of Technetium-99m will be remaining is
T((d) 10 %) =100×[tex](\frac{1}{2} )^{3.322}[/tex]
