# Counting days involving half-life

#### Estermont

On 31st of August, Harry is left with 100 kg of a certain element comprised of isotope A (40%, half-life: 1 day) and isotope B (60%, half-life: 3 days) after putting it through a centrifuge. What mass of the element will be left on 15th September to the nearest g? (Half-life is the amount of time it takes for half of the original substance to decay)

My only problem with this question is counting the number of days... Is it 15 or 16? And is my method for this question correct?

100 x 0.4 = 40 kg of Isotope A
100 x 0.6 = 60 kg of Isotope B

If we say the number of days between August 31st and September 15th is 15,

15/1 = 15
15/3 = 5

40 x 0.515= 5/4096 kg
60 x 0.55= 1.875 kg

1.2207... + 1875 = 1876 g

#### Cervesa

Aug 31 is day 0
Sep 1 is day 1
.
.
.
Sep 15 is day 15

Half-life indicates the substance decays; you end up with less than you started.

$A_0 = 40$

$A(t) = 40 \cdot \left(\dfrac{1}{2}\right)^t \implies A(15) = 40 \cdot \left(\dfrac{1}{2}\right)^{15} = \dfrac{5}{2^{12}}$

$B_0 = 60$

$B(t) = 60 \cdot \left(\dfrac{1}{2}\right)^{t/3}$

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