2 math questions! please help!?

Question

i'm confused. please help. how would i begin to solve these?

1) the half life of phosphorus-32 is about 14 days. there are 6.6 grams present initally.

a) express the amount of phosphorus-32 remaining as a function of time t.

b) when will there be 1 gram remaining?

2) suppose in any given year, the number of cases of a disease is reduced by 20%. if there are 10,000 cases today, how many years will it take

a) to reduce the number of cases to 1000

b) to eliminate the disease; that is, to reduce the number of cases to less than 1

the help is much appreciated!

Answer 1

1)The formula for 1/2 life is

m(t) = m(0) * (.5 ^ (t/t_half)) ...so

m(t) = 6.6 (.5^(t/14))

1 gram remaining is:

1 = 6.6(.5^(t/14)).....divide by 6.6

.1515 = .5^t/14.....log both sides

log .1515 = t/14 log .5.....divide by log .5

log .1515 / log .5 = t/14.....simplify left side

2.72 = t/14.....multiply by 14

38.1 = t

2)The formula will be:

10000 * e^-.2t so...

10000 * e^-.2t = 1000.....divide by 10000

e^-.2t = .1.....take ln of each side

-.2t = -2.3.....divide by -2

t = 11.5 years

10000 * e^-.2t < 1.....divide by 10000

e^-.2t < .0001.....ln both sides

-.2t < -9.21.....divide by -2

t > 46, more than 46 years