1. carbon-14 is used to determine the age of artifacts in carbon dating. carbon-14 has a half-life of 5730 years. write an exponential decay function for a 52-mg sample

See the attached graphic.  (I'm rewording the attached formula)
Ending amount = Beginning Amount * e ^(k * t)
where "e" is the mathematical constant 2.718281828
and k = natural log (.5) / half-life
k = -.693147 / 5,730 = -0.000120968062827225
Ending amount = 52 * e ^(-0.000120968062827225* t)

As a test, let's see how much remains after 11,460 years.
Ending amount = 52 * e ^(-0.000120968062827225 * 11,460)
Ending amount = 52 * e ^ ( -1.386294 )
Ending amount = 52 * 0.250000
Ending amount = 13
(which is correct because that is one quarter of the amount after two half-lives).

Source:
http://www.1728.org/halflif2.htm

You didn’t put the necessary image to solve the question : p