A wooden artifact from an ancient tomb contains 70% of the carbon-14 that is present in living trees. How long ago was the artifact made? (The half-life of carbon-14 is 5730 years. Round your answer to the nearest whole number.)

And....

An infectious strain of bacteria increases in number at a relative growth rate of 200% per hour. When a certain critical number of bacteria are present in the bloodstream, a person becomes ill. If a single bacterium infects a person, the critical level is reached in 24 hours. How long will it take for the critical level to be reached if the same person is infected with 9 bacteria? (Round your answer to two decimal places.)

the answer to ther second one is 22.85

#1

70%=0.7 (because70%/100%=0.7)
The half-life 1/2=0.5

Equation: 0.7=0.5^(t/5730)

t=5730log0.7/log0.5=2949(years)

To answer the first question, we can use the concept of a half-life to determine how long ago the wooden artifact was made.

The half-life of carbon-14 is the time it takes for half of the radioactive carbon-14 atoms to decay. In this case, since the artifact contains 70% of the carbon-14 that is present in living trees, we can assume that it has undergone one half-life.

To determine the age of the artifact, we can set up an equation:

0.5^x = 0.7

Where x represents the number of half-lives that have occurred since the artifact was made.

To solve for x, we can take the logarithm (base 0.5) of both sides of the equation:

log(0.5^x) = log(0.7)
x*log(0.5) = log(0.7)
x = log(0.7) / log(0.5)

Using a calculator, we can find that x is approximately 0.51457.

Since x represents the number of half-lives, we can multiply it by the half-life of carbon-14 (5730 years) to find the age of the artifact:

Age = 0.51457 * 5730 ≈ 2945.6 years

Rounded to the nearest whole number, the artifact was made approximately 2946 years ago.

Moving on to the second question, we can use the concept of exponential growth to determine the time it takes for the critical level of bacteria to be reached.

The relative growth rate of 200% per hour means that the number of bacteria doubles every hour.

If it takes 24 hours for a single bacterium to reach the critical level, we can set up an equation:

2^x = 9

Where x represents the number of hours it takes for the critical level to be reached when starting with 9 bacteria.

To solve for x, we can take the logarithm (base 2) of both sides of the equation:

log(2^x) = log(9)
x*log(2) = log(9)
x = log(9) / log(2)

Using a calculator, we can find that x is approximately 3.16993.

Therefore, it will take approximately 3.17 hours for the critical level to be reached when starting with 9 bacteria, rounded to two decimal places.