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 of P-32 present after t days, we can use the formula for exponential decay:
Q(t) = Q0 * e^(-kt),
Where:
Q(t) is the amount present after t days,
Q0 is the initial amount (200g in this case),
e is the base of the natural logarithm (approximately 2.71828),
k is the decay constant that depends on the half-life,
t is the time in days.
To determine the decay constant (k), we can use the formula:
k = ln(2) / t_(1/2),
Where:
ln(2) is the natural logarithm of 2,
t_(1/2) is the half-life.
Given that the half-life of P-32 is 14.2 days, we can calculate the decay constant:
k = ln(2) / 14.2 ≈ 0.0488 (rounded to four decimal places).
Now, we can calculate the amount Q(t) present after t days:
Q(t) = 200 * e^(-0.0488t).
To find how fast P-32 is decaying at a specific time t = 21.5, we need to find the derivative of Q(t) with respect to t, and then substitute t = 21.5 into the derivative equation.
Let's find the derivative:
dQ(t)/dt = -0.0488 * 200 * e^(-0.0488t).
Now, substitute t = 21.5 into the equation:
dQ(t)/dt = -0.0488 * 200 * e^(-0.0488 * 21.5).
Calculating this expression will provide the answer for how fast P-32 is decaying at t = 21.5.