Answer:
The density of the substance when it was deposited 55 years ago was 60.54 mg/cm³
Step-by-step explanation:
The exponential function for growth and decay is,
[tex]y(t)=a(1\pm r)^t[/tex]
where,
y(t) = the amount after time t
a = initial amount
r = rate of change
t = time period
+ is used for growth and - is used for decay.
As this is the case of decay, so the function becomes,
[tex]y(t)=a(1- r)^t[/tex]
Given,
y(55) = 9 mg/cm³
r = 50% = 0.5 (as the substance is getting halved)
t = [tex]\dfrac{55}{20}[/tex] = 2.75 (as the half life is 20 years and we have convert time in terms of half life)
Putting the values,
[tex]\Rightarrow 9=a(1- 0.5)^{2.75}[/tex]
[tex]\Rightarrow 9=a(0.5)^{2.75}[/tex]
[tex]\Rightarrow a=\dfrac{9}{(0.5)^{2.75}}[/tex]
[tex]\Rightarrow a=60.54\ mg/cm^3[/tex]
