A radioactive material disintegrates at a rate proportional to the amount currently present. If Q(t) is the amount present at time t, then dQ/dt =−rQ where r>0 is the decay rate. Required:
If 100 mg of a mystery substance decays to 81.14 mg in 4 weeks, find the time required for the substance to decay to one-half its original amount.

For the substance to decay to one-half of its original amount, it will take:

10.6 weeks.

Step-by-step explanation:

The decay rate = dQ/dt= -rQ

Where r > 0.

100 mg of a mystery substance decays to 81.14 mg in 4 weeks

This implies that it has decayed by 18.86 mg (100 - 81.14) in 4 weeks

The rate of decay per week = 4.715 mg (18.86 mg/4)

For the substance to decay to one-half of its original amount, i.e. 50 mg (100 mg/2), it will take 10.6 (4/18.86 * 50) weeks.

We can also determine the time it will take the substance to decay by working with the decay rate and the volume of decay, thus:

Decay rate = 4.715 / week

Decay volume = 50 mg (100 mg/2)

Time for 50 mg to decay = Decay volume/Decay rate

= 50 mg/4.715

= 10.6 weeks

ddd

step-by-step explanation: