In the year 2005, a picture supposedly painted by a famous artist some time after 1815 but before 1865
contains 98.4 percent of its carbon-14 (half-life 5730 years).
(a) From this information, could this picture have been painted by this artist?
(b) Approximately how old is the painting
you know that the fraction of C14 after t years is
(1/2)^(t/5730)
So, now you just need to find t such that
(1/2)^(t/5730) = 0.984
and check to see whether that number of years fits the time period specified.
To determine whether the picture could have been painted by the artist and estimate its age, we can use the concept of carbon-14 dating.
(a) Carbon-14 dating can only be reliably used on organic materials that were once alive, such as wood or plant fibers used in paint. If the picture was painted by the artist, it would have likely used organic materials, making carbon-14 dating applicable. However, it's important to note that carbon-14 dating alone cannot confirm or attribute the painting to a specific artist.
(b) The half-life of carbon-14 is 5730 years, meaning that after each half-life, the remaining amount of carbon-14 is reduced by half. To estimate the age of the painting, we can use the following formula:
Age of the painting = (ln(Ratio of carbon-14 in the painting to the initial amount of carbon-14))/(ln(1/2)) * Half-life of carbon-14
In this case, the ratio of carbon-14 remaining is 98.4% or 0.984, and the half-life is 5730 years.
Age of the painting = (ln(0.984))/(ln(1/2)) * 5730
Simplifying this equation, we get:
Age of the painting ≈ (0.016)/(ln(2)) * 5730
Calculating this approximation:
Age of the painting ≈ 11,460 years
Therefore, the painting is approximately 11,460 years old.
To answer these questions, we need to understand the concept of carbon-14 dating and its application in determining the age of organic materials.
Carbon-14 dating relies on the fact that all living organisms contain a small amount of carbon-14, a radioactive isotope of carbon. When an organism dies, it stops taking in new carbon-14, and the existing carbon-14 starts to decay at a predictable rate, known as its half-life. The half-life of carbon-14 is approximately 5730 years.
Now, let's tackle each question one by one:
(a) Could this picture have been painted by this artist?
To determine if the picture could have been painted by this artist, we need to check if the carbon-14 concentration aligns with the timeframe in which the artist was active. The fact that the painting contains 98.4 percent of its carbon-14 suggests it is relatively recent and doesn't significantly predate the time in which the artist lived (after 1815 but before 1865). Based on this information alone, it is plausible that the picture could have been painted by this artist.
(b) Approximately how old is the painting?
To estimate the age of the painting, we can use the concept of carbon-14's half-life. Since the painting contains 98.4 percent of its carbon-14, it means that only 1.6 percent has decayed. We can calculate the number of half-lives that have passed to reach this percentage.
First, we need to convert the percentage into a fraction: 1.6 percent = 0.016.
To calculate the number of half-lives, we can use the formula: number of half-lives = (ln(initial fraction of carbon-14) / ln(0.5)).
Taking the natural logarithm (ln) of 0.016 and dividing it by the ln of 0.5, we can find the number of half-lives.
number of half-lives = (ln(0.016) / ln(0.5)) ≈ 4.11
Hence, approximately 4.11 half-lives have passed since the painting was created.
Given that the half-life of carbon-14 is approximately 5730 years, we can calculate the approximate age of the painting using the formula: age = (number of half-lives * half-life).
age ≈ 4.11 * 5730 ≈ 23,571 years
Therefore, the approximate age of the painting is around 23,571 years.
However, it is worth noting that the calculated age seems inconsistent with the information provided, as it falls outside the time when the artist supposedly painted it. Therefore, based on this analysis, it is unlikely that the painting was actually created by this artist.