thirty percent of a radioactive substance decays in four years. assuming the decay is exponential, find the half life of the substance.

the beginning of 1975 the population of a country was 40 million and growing at a rate of 3% per year. Assume that the growth is exponential. Estimate the population of the country at the beginning of the year 2010

the decay function after t years is

0.3^(t/4)
You want the half-life, or
0.5^(t/k)

0.5^(t/k) = 0.3^(t/4)
t/k log 0.5 = t/4 log 0.3
k = 4log.5/log.3 = 2.3

so, you have (1/2)^(t/2.3)
and thus the half-life is 2.3 years

for the other, the population is

40*1.03^t

To find the half-life of a radioactive substance when the decay is exponential, we can use the formula:

t1/2 = (ln 2) / k

Where:
t1/2 = half-life
k = decay constant

Given that 30% of the substance decays in four years, we can find the decay constant (k) using the formula:

0.30 = e^(-4k)

Taking the natural logarithm of both sides:

ln 0.30 = -4k

Solving for k:

k = (ln 0.30) / (-4)

Now, we can substitute the value of k into the formula for half-life:

t1/2 = (ln 2) / k
t1/2 = (ln 2) / [(ln 0.30) / (-4)]

Calculating the approximate value of t1/2:

t1/2 = (ln 2) / [(ln 0.30) / (-4)]

This calculation will give you the half-life of the substance.

Regarding the second part of your question:

To estimate the population of a country at the beginning of the year 2010, given an initial population of 40 million in 1975 and a growth rate of 3% per year assumed to be exponential, we can use the formula for exponential growth:

P(t) = P0 * e^(rt)

Where:
P(t) = population at time t
P0 = initial population
r = growth rate (in decimal form)
t = time

Given the information, we can substitute the values into the formula:

P(t) = 40 million * e^(0.03 * t)

To estimate the population at the beginning of 2010, we need to find the value of t when t = 2010 - 1975 = 35 years.

P(t) = 40 million * e^(0.03 * 35)

Calculating this expression will give you an estimate of the population at the beginning of the year 2010.

To find the half-life of a radioactive substance that decays exponentially, we can use the following formula:

N(t) = N0 * (1/2)^(t / T)

Where:
N(t) is the amount of substance remaining after time t
N0 is the initial amount of substance
T is the half-life of the substance

In this case, we're given that 30% of the substance decays in four years. So, the remaining amount of substance after four years is 70% or 0.7. Plugging these values into the formula, we get:

0.7 = 1 * (1/2)^(4 / T)

To solve for T, we need to isolate T on one side of the equation. Taking the logarithm of both sides with base 1/2, we get:

log(0.7) = log((1/2)^(4 / T))

Using the logarithmic property log(a^b) = b * log(a), the equation becomes:

log(0.7) = (4 / T) * log(1/2)

Now, we can solve for T by rearranging the equation:

T = (4* log(1/2)) / log(0.7)

Using a calculator, we can evaluate this expression to find the value of T, which represents the half-life of the radioactive substance.

For the second question, we're given that the population of a country at the beginning of 1975 was 40 million and growing at a rate of 3% per year. Assuming exponential growth, we can use the formula:

P(t) = P0 * (1 + r)^t

Where:
P(t) is the population after time t
P0 is the initial population
r is the growth rate as a decimal
t is the time in years

In this case, we want to estimate the population at the beginning of the year 2010, which is 35 years after 1975. Plugging in the given values, we have:

P(35) = 40 million * (1 + 0.03)^35

Using a calculator, we can evaluate this expression to estimate the population at the beginning of 2010.