Suppose an unknown radioactive substance produces 8000 counts per minute on a Geiger counter at a certain time, and only 500 counts per minute 13 days later. Assuming that the amount of radioactive substance is proportional to the number of counts per minute, determine the half-life of the radioactive substance.

The half-life of the radioactive substance is of 3.25 days.

Step-by-step explanation:

The amount of radioactive substance is proportional to the number of counts per minute:

This means that the amount is given by the following differential equation:

In which k is the decay rate.

The solution is:

In which Q(0) is the initial amount:

8000 counts per minute on a Geiger counter at a certain time

This means that

500 counts per minute 13 days later.

This means that . We use this to find k.

So

Determine the half-life of the radioactive substance.

This is t for which Q(t) = 0.5Q(0). So

The half-life of the radioactive substance is of 3.25 days.

x^6

step-by-step explanation:

this is correct because:

x-6=0

0*x=0

so

x^6

b. 6

step-by-step explanation:

we are given the table representing an exponential function.

we can see from the table that, as x is increasing, the value of y is increasing in the power of 6.

i.e. the following relation is obtained,

x = 1, y = 6

x = 2, y = 36 =

x = 3, y = 216 =

x = 4, y = 1296 =

x = 5, y = 7776 =

so, the geometric series obtained is .

thus, the common ratio is given by = = 6.

hence, the common ratio is 6.