Question

Radioactive Decay The half-life of a certain radioactive substance is 12 hours. There are 8 grams present initially. (a) Express the amount of substance remaining as a function of time $t . (b) When will there be 1 gram remaining?

Short answer

The function expressing the amount of substance remaining as a function of time is \( Q(t) = 8 * (1/2)^{t/12} \). The time when there will be 1 gram remaining is approximately found by solving \( t = 12 * ln(1/8) / ln(1/2) \) hours.

Step-by-step solution

Set up the function for the amount of substance

To start, use the given information to setup the decay function. The initial amount of the substance is 8 grams and the half-life is 12 hours. Plug these values into the exponential decay formula to obtain the function \( Q(t) = 8 * (1/2)^{t/12} \).

Extrapolate the amount of substance for any given time

With the decay function setup, one can extrapolate the amount of substance remaining for any given time t. Simply input the designated hours into the decay function and solve for the substance amount.

Determine the time for 1 gram remaining

For the second part of the problem, we need to determine the time when there's only 1 gram of substance remaining. Set up the equation to solve for \( t \) yielding \( 1 = 8 * (1/2)^{t/12} \). Solve this equation for \( t \). To do this, apply logarithm on both sides of the equation and then use properties of logarithm and exponential functions.

Solving the equation for t

Solving the equation in Step 3, In the equation \( 1 = 8 * (1/2)^{t/12} \), divide both sides by 8 to obtain \(1/8 = (1/2)^{t/12} \). Then, apply the natural logarithm ln(.) to both sides of the equation to obtain ln(1/8) = \( t/12 \) * ln(1/2). Solving for \( t \), \( t = 12 * ln(1/8) / ln(1/2) \), gives us the number of hours when 1 gram is remaining.