Math help I forget how to do these...?

Question

Suppose the half-life of a certain radioactive substance is 20 days and that there are 5 grams present initially. When will there be only 1 gram of the substance remaining?

The population of Knoxville is 500,000 and is increasing at the rate of 3.75% each year. Approximately when will the population double?

Answer 1

Problem #1

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Given these definitions:

m(t) = mass at time "t"

t = time passed

h = half life

The formula for half life is:

m(t) = m(0) * 0.5^(t/h)

Substituting in the values we know and solving for "t":

1 = 5 * 0.5^(t/20)

1/5 =  0.5^(t/20)

ln(1/5) = ln(0.5^(t/20))

ln(1/5) = (t/20) * ln(0.5)

ln(1/5)/ln(0.5) = (t/20)

20* ln(1/5)/ln(0.5) = t

20 * (-1.609437912)/(-0.693147181) = t

46.438562 = t

There will be 1 gram of the substance remaining in

about 46.4 days.

To verify, use the calculator at:

http://www.1728.com/halflife.htm

Problem #2

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Given this population formula:

p(t) = p(0)* (1.0375)^t

Solve for "t":

1000000 = 500000*(1.0375)^t

1000000/500000 = (1.0375)^t

2 = (1.0375)^t

ln(2) = ln((1.0375)^t)

ln(2) = t*ln(1.0375)

ln(2)/ln(1.0375) = t

0.693147181 / 0.036813973 = t

18.8283721 = t

So the population will double (reach 1 million) in

approximately 18.8 years.

As a quick check, put the numbers into a spreadsheet:

0    500000.00

1    518750.00

2    538203.12

3    558385.74

4    579325.21

5    601049.90

6    623589.27

7    646973.87

8    671235.39

9    696406.72

10   722521.97

11   749616.55

12   777727.17

13   806891.93

14   837150.38

15   868543.52

16   901113.90

17   934905.67

18   969964.64

19  1006338.31

Notice that by 19 years, the population has exceeded

1 million, so the calculation is correct.

.