Plutonium-238 is used in bombs and power plants but is dangerously radioactive. It decays very slowly into radioactive materials. If you started with 300 grams today, a year from now you would still have 297.6 grams.
Construct an exponential function to describe the DECAY of plutonium-238 over time, rounding to three decimal places if necessary.
P(t)=________
P(t) = 300*P*e^(-k*t)
t is in years
k = .011398045
1. Diff eq is dP/dt = -k*P
2. General soln is P = P0*e^(-kt)+1
3. Find k by letting P = 297.6 and P0=300
There are a few steps between 1 and 2 here. Well beyond college algebra.
To construct an exponential function to describe the decay of plutonium-238 over time, we can use the exponential decay model. The general formula for exponential decay is:
P(t) = P(0) * e^(-kt)
Where:
P(t) is the amount of plutonium-238 remaining at time t
P(0) is the initial amount of plutonium-238
e is Euler's number (approximately equal to 2.71828)
k is the decay constant
Given that the initial amount of plutonium-238 is 300 grams, and after one year it reduces to 297.6 grams, we can substitute these values into the equation to solve for the decay constant (k).
297.6 = 300 * e^(-k * 1)
Divide both sides by 300:
0.992 = e^(-k)
To isolate the exponent, take the natural logarithm (ln) of both sides:
ln(0.992) = ln(e^(-k))
ln(0.992) = -k
Now we can solve for k by calculating the natural logarithm of 0.992:
k ≈ -ln(0.992)
Using a calculator or mathematical software, we find that k is approximately -0.008.
Now we can construct the exponential function for the decay of plutonium-238:
P(t) = 300 * e^(-0.008t)
Rounding to three decimal places, the exponential decay function describing the decay of plutonium-238 over time is:
P(t) ≈ 300 * e^(-0.008t)