Answer:
t = 39.42 days
Step-by-step explanation:
To do this, we need to use the general expression for decay rate:
R = R₀ exp(-λt) (1)
According to the problem, decay rate of A initially is 64 times rate of B, so we can say the following:
A₀ = 64B₀ (2)
The value of λ can be determined using:
λ = ln2 / t(1/2) (3)
Let's calculate first the value of λ for each nuclei:
For A: λ₁ = ln2 / 5 = 0.1286 day⁻¹
For B: λ₂ = ln2 / 30 = 0.0231 day⁻¹
Now, let's write an expression using (1) for each nuclei.
A = A₀ exp(-0.1286t)
A = 64B₀ exp(-0.1286t) (3)
B = B₀ exp(-0.0231t) (4)
We want to know the time when A = B , therefore, we can actually equals (3) and (4) and solve for t:
64B₀ exp(-0.1286t) = B₀ exp(-0.0231t)
64 exp(-0.1286t) = exp(-0.0231t)
ln(64 * exp(-0.1286t)) = ln(exp(-0.0231t))
ln64 + ln(exp(-0.1286t))= -0.0231t
4.1589 - 0.1286t = -0.0231t
4.1589 = (0.1286 - 0.0231)t
t = 4.1589 / 0.1055
t = 39.42 days
So, in 39 days, the decay rates of A and B will be the same.
Hope this helps