A given substance decays at a rate proportional to the square of the amount of the material present. At the end of one year there is 0.5 grams of the substance remaining.
- Create and solve a differential equation that governs the mass of the material present after years.
- Find the decay constant of the material in units .
From our given information, we know that the rate of decay is proportional to the square of the amount present at time t. If we let y(t) denote how much material is there at time t, our differential equation becomes
Where k is a constant
The above differential equation is separable, so we can turn the equation into the following
Integrating from time 0 to some arbitrary time x, we get
Now, we also know from our given information that y(1) = 1/2 g. So, we can find in terms of y(0) as follows:
Plugging this value into our equation for y gives us:
And once again using our given value y(1) = 1/2
Now we can plug this value back into our equations for k and y
A side note: I had to find the value of k before finding the function y, which is in reverse from the way the book presented the problem. Nevertheless, the answers are as requested.