half-life

Examples of half-life in the following topics:

• Exponential Decay

  • As an example let us assume we have a $100$ pounds of a substance with a half-life of $5$ years.
  • Half-life is very useful in determining the age of historical artifacts through a process known as carbon dating.
  • C-13 has a half-life of 5700 years—that is, in 5700 years, half of a sample of C-13 will have converted to C-12, which represents approximately all the remaining carbon.
  • Using the graph, find that half-life.
  • Since there is 50% of the substance left after 1 year, the half-life is 1 year.

• Natural Logarithms

  • For example, the doubling time for a population which is growing exponentially is usually given as ${\ln 2 \over k}$ where $k$ is the growth rate, and the half-life of a radioactive substance is usually given as ${\ln 2 \over \lambda}$ where $\lambda$ is the decay constant.

• Problem-Solving

  • According to the United States Census Bureau, over the last 100 years (1910 to 2010), the population of the United States of America is exponentially increasing at an average rate of one and a half percent a year (1.5%).
  • A related constant is half-life, which is the amount of time required for a process to consume half the original quantity, or the time it takes a decaying substance to drop to half its original value.

• Ellipses

  • How often do ellipses come up in real life?
  • The semi-major axis (denoted by a in the figure) and the semi-minor axis (denoted by b in the figure) are one half of the major and minor axes, respectively.
  • The area enclosed by an ellipse is $\pi ab$, where a and b are one-half of the ellipse's major and minor axes respectively.

• Graphing Inequalities

  • For a linear equality in two variables, all solutions are located in one entire half-plane.
  • Next, choose a test point to figure out which half-plane we need to shade in.
  • This a true statement, so shade in the half-plane containing $(0, 0).$
  • All points in the shaded half-plane above the line are solutions to this inequality.
  • The boundary line shown above divides the coordinate plane into two half-planes.

• Double and Half Angle Formulae

  • Trigonometric expressions can be simplified by applying the double- and half-angle formulae.
  • The half-angle formulae can be derived from the double-angle formulae.
  • They are useful for finding the trigonometric function of an angle $\theta$ which is half of a special angle $\alpha$ (in other words, $\displaystyle{\theta = \frac{\alpha}{2}}$).
  • The half-angle formulae are as follows:
  • We can thus apply the half-angle formula with $\alpha = 30^{\circ}$:

• Circles as Conic Sections

  • You've known all your life what a circle looks like.
  • Radius: a line segment joining the center of the circle to any point on the circle itself; or the length of such a segment, which is half a diameter.

• Functions and Their Notation

  • Let's say the machine has a blade that slices whatever you put in into two and sends one half of that object out the other end.
  • If you put in a banana you would get back half a banana.
  • If you put in an apple you would get back half an apple.
  • Let's define the function to take what you put into it and cut it in half.
  • This shows a function that takes a fruit as input and releases half the fruit as output.

• Interval Notation

  • An interval that has only one real-number endpoint is said to be half-bounded, or more descriptively, left-bounded or right-bounded.
  • For example, the interval $(1, + \infty)$ is half-bounded; specifically, it is left-bounded.

• Graphing Quadratic Equations in Vertex Form

  • When this is the case, we look at the coefficient on $x$ (the one we call $b$) and take half of it.
  • Note that half of $6$ is $3$ and $3^2=9$.