Phosphorus-32 (P-32) has a half-life of 14.2 days. If 200 g of this substance are present initially, find the amount Q(t) present after t days. (Round your growth constant to four decimal places.)
How fast is the P-32 decaying when t = 21.5? (Round your answer to three decimal places.)
To find the amount Q(t) of Phosphorus-32 (P-32) present after t days, we can use the formula:
Q(t) = Q(0) * exp(-k*t)
Where:
Q(t) is the amount of P-32 present after t days
Q(0) is the initial amount of P-32, which is 200 g in this case
exp is the exponential function
k is the decay constant, which can be found using the formula:
k = ln(2) / half-life
So, let's calculate k first:
k = ln(2) / 14.2 days
k ≈ 0.0488 (rounded to four decimal places)
Now we can find the amount Q(t) present after t days:
Q(t) = 200 * exp(-0.0488*t)
To find how fast P-32 is decaying at a specific time t = 21.5 days, we need to find the derivative of Q(t) with respect to t. In other words, find dQ/dt.
dQ/dt = -k * Q(0) * exp(-k*t)
Now, let's substitute the values:
dQ/dt = -0.0488 * 200 * exp(-0.0488 * 21.5)
Calculating this expression will give us the rate at which P-32 is decaying at t = 21.5 days.