SOLUTION: The rate of decay of radium is said to be proportional to the amount of radium present. If the half-life of radium is 1700 years and there are 400 grams on hand now, how much rad

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons  -> Quadratic Equation Lesson -> SOLUTION: The rate of decay of radium is said to be proportional to the amount of radium present. If the half-life of radium is 1700 years and there are 400 grams on hand now, how much rad      Log On


   

Question 1194522: The rate of decay of radium is said to be proportional to the amount of radium
present. If the half-life of radium is 1700 years and there are 400 grams on hand now, how
much radium will be present in 900 years given the exponential decay equation, 𝑦 = Ce^kt.
what is the value of e^k, and how much radium will be present in 845 years.
Found 2 solutions by josgarithmetic, Theo:
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
Half-life information, 1%2Ae%5E%28kt%29=1%2F2
ln%281%29%2Bln%28e%5E%28kt%29%29=ln%281%2F2%29
kt=ln%281%2F2%29
kt=-ln%282%29
k=%28-ln%282%29%29%2Ft
k=%28-ln%282%29%29%2F1700
k=-0.0004077

highlight_green%28y=Ce%5E%28-0.0004077t%29%29

Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
formula is f = p * e ^ (kt)
half life formula is 1/2 = e ^ (1700k)
take the natural log of both sides of the equation to get:
ln(1/2) = ln(e^1700k)
this becomes:
ln(1/2) = 1700k * ln(e) which becomes ln(1/2) = 1700k
solve for k to get:
k = ln(1/2)/1700 = -4.07733636 * 10^-4
confirm by replacing k in the original equation to get:
f = p * e ^ (-4.07733636 * 10 ^ -4 * 1700) = .5
this confirms the value of k is correct.
if the present value is 400, the value will be 200 in 1700 years.
f = 400 * e ^ ((-4.07733636 * 10 ^ -4 * 1700)) = 200.
value of k is confirmed again.
in 900 years, the value will be 400 * e ^ (-4.07733636 * 10 ^ -4 * 900) = 277.1348678.
in 845 years, the value will be 400 * e ^ (-4.07733636 * 10 ^ -4 * 845) = 283.4199231.
the value of e^k is equal to e^ (-4.07733636 * 10 ^ -4) = .9995923495.
not sure why you'd want this because the formula required is e ^ (kt).
e^k means that t = 1.
i graphed the equation.
the equation graphed is y = e ^ (-4.07733636 * 10 ^ -4 * x)
it is shown below.

let me know if you have any questions.
theo