Question

The Half-life of a certain substance is 20 mins. You start with 45 mg of the substance

Answer 1

a. Remaining amount after t minutes = 45 * $(1/2)^{\frac{t}{20 }$

b.  0.010986 mg

c. 109.84 minutes

Step-by-step explanation:

Step 1:

The decay of the substance is exponential. It is given that half life is 20 minutes and initially we start with 45 mg. So in 20 minutes we will have remaining amount 45/2, in 40 minutes remaining amount will be 45/4 etc.

This can be modeled by the equation:

Remaining amount after t minutes = Original amount * $(1/2)^{\frac{t}{t_{1/2} }$

where  $t_{\frac{1}{2}}$ represents the half life.

The half life for this substance is 20 minutes.

Hence the Remaining amount after t minutes = 45 * $(1/2)^{\frac{t}{20 }$

Step 2:

We need to calculate the amount remaining after 4 hours (240 minutes).

Substituting t = 240 in above equation we get

Amount remaining after 4 hours = 45 * $(1/2)^{\frac{240}{20 }$ = 0.010986 mg

Step 3:

We need to calculate the time taken when 1 milligram is left. Substituting in the equation we get

1 = 45 * $(1/2)^{\frac{t}{20 }$  

Taking log on both sides we get

t = $\frac{20 ln (1/45)}{ln 1/2}$ = 109.84 minutes

Time taken for 1 mg of the substance to be left = 109.84 minutes

Step 4 :

Answer :

a. Remaining amount after t minutes = 45 * $(1/2)^{\frac{t}{20 }$

b.  0.010986 mg

c. 109.84 minutes