A specific radioactive substance follows a continuous exponential decay model. It has a half-life of minutes. At the start of the experiment, 82.6g is present.

a. Let t be the time (in hours) since the start of the experiment, and let y be the amount of the substance at time t. Write a formula relating y to t. Use exact expressions to fill in the missing parts of the formula.
Do not use approximations.

y= _e^____

b. How much will be present in 18 hours?

Half life of a substance is the time the substance to reduce to half of its original. Half life is expressed as [tex]t_{1/2} = \dfrac{0.693}{\lambda}[/tex] where [tex]\lambda[/tex] is the decay constant.

Given half life of the substance to be 12 hours, we can get the decay constant

[tex]t_{1/2} = \dfrac{0.693}{\lambda}\\12 = \dfrac{0.693}{\lambda}\\\lambda = \dfrac{0.693}{12}\\\lambda = 0.05775hr^{-1}[/tex]

a) Given the initial mass of the substance y₀ = 82.6g, we can find the amount of substance remaining after time t using the formula;

[tex]y(t) = y_0 e^{-\lambda t}[/tex]

Substituting y₀ = 82.6g and λ = 0.05775 into the formula given;

[tex]y(t) = 82.6 e^{-0.05775 t}[/tex]

Hence the formula relating y to t is expressed as [tex]y(t) = 82.6 e^{-0.05775 t}[/tex]

b) To calculate the amount that will be present after 18 hours, we will substitute t = 18 into the expression in (a) as shown;

[tex]y(t) = 82.6 e^{-0.05775 t}\\y(t) = 82.6 e^{-0.05775 *18}\\y(t) = 82.6 e^{-1.0395}\\y(t) = 82.6 * 0.35363\\y(t) = 29.2099958g[/tex]

Hence about 29.2099958 g will be present after 18 hours