Answer:
Carbon/Radiocarbon Dating
Explanation:
Carbon dating yields some of the most accurate results for aging samples around 50,000 years old, and can be as accurate as within a few decades of the exact year. Carbon dating uses the relative proportions of the carbon isotopes carbon-12 and carbon-14 that the fossil/sample contains to pinpoint an almost exact time.
(a)
We have been given an equation
. Upon taking derivative of this equation with respect to x on both the sides, we get:

(b)
In order to write the equation of tangent line, we need to find the slope of the line. We know that point (3,11) lies on the graph. Therefore, we write the slope of the tangent as:

Therefore, equation of tangent line is:

(c)
We know that slope of tangent is given as
. The tangent will be vertical when denominator of slope is zero, that is:

Answer:
The subjects of a conquered kingdom are not used to living in freedom or standing up for themselves, and are therefore slower to take up arms than are the citizens of a republic. There are three options for the conquering prince: destroy the republic, live there, or set up a puppet government. At the end of the day, Machiavelli seems to favor the first option.