only 3 Calculus question regarding population, Please help?

Question

The answers I’m getting are different. Can someone please help me get the right answer? Thanks a lot
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2. The population of US was 3.9 million in 1790 and 178 million in 1960. if the rate of growth is assumed proportional to the number present, what estimate would you give for the population in 2000? (Compare your answer with the actual 2000 population, which is 275 million?)
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4. Human hair from a grave in Africa proved to have only 51% of the carbon 14 of living tissue. When was the body buried? See problem 5)
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5. (carbon Dating) all living things contain carbon 12, which is stable, and carbon 14, which is radioactive. While a plant or animals is alive, the ratio of these two isotopes of carbon remains unchanged, sinc

Answer 1

2.) This is a simple exponential question:
P = Pi * e^(kt) where k is the exponential constant you need to find.

Part 1: Find K given the 2 given years and populations.
P = 178 * 10^6
Pi = 3.9 * 10^6
t = 170 years
178 * 10^6 = 3.9 * 10^6 * e^(k * 170)
178 * 10^6 / 3.9 * 10^6 = e^(k * 170)
45.641 = e^(k * 170)
ln 45.641 = k * 170 [remember that ln (e^x) = x]
k = ln 45.641 / 170 = 0.022475

Part 2: Use the population in 1960 and k to find the population in 2000.
P =
Pi = 178 * 10^6
t = 40 years
P = 178 * 10^6 * e^(0.022475 * 40)
P = 437 million

The number is of course way off, because it does not account for changes in birth, death, and immigration rates.

4. and 5.) Your question for #5 is cut off, however, I can answer generally how to handle these questions.

Carbon-14 has a half-life of 5730 years. Thus, a sample that has 50% of the normal ratio of carbon-14 is 5730 years old.

For #4, the equation for radioactive decay is:
N = Ni * 2^(-t/λ)
where λ = the half life of C-14
Ni is the initial amount of C-14
N is the current amount of C-14.

As given in the problem N / Ni = 51%, so the resulting equation is:
.51 = 2^(-t / 5730)
log .51 = (-t / 5730) * log 2
log .51 / log 2 = (-t / 5730)
-0.9714 * 5730 = -t
-5566.3 = -t
t = 5566.3 years, rounded to 5560 years.