doubling time

Examples of doubling time in the following topics:

• 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

  • With this function, you can figure out how many organisms you will have after a certain amount of time.
  • This means that the doubling time of the American population (depending on the yearly growth in population) is approximately 50 years.
  • 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.
  • For instance, in the microorganisms example, if the current value is 8, then the next time step will give 16, whereas if the current level is 1024, the next time step gives you 2048.
  • In this case, the y-axis can be taken as the bacterial count, and the x-axis is time.

• Direct and Inverse Variation

  • If $x$ and $y$ are in direct variation, and $x$ is doubled, then $y$ would also be doubled.
  • For example, doubling $y$ would result in the doubling of $x$.
  • For example, if $x$ and $y$ are inversely proportional, if $x$ is doubled, then $y$ is halved.
  • As an example, the time taken for a journey is inversely proportional to the speed of travel.

• Applications of Hyperbolas

  • A hyperbola is an open curve with two branches and a cut through both halves of a double cone, which is not necessarily parallel to the cone's axis.
  • As we should know by now, a hyperbola is an open curve with two branches, the intersection of a plane with both halves of a double cone.
  • At most populated latitudes and at most times of the year, this conic section is a hyperbola.
  • A ship can locate its position using the arrival times of signals from GPS transmitters.
  • A hyperbola is an open curve with two branches, the intersection of a plane with both halves of a double cone.

• Double and Half Angle Formulae

  • Trigonometric expressions can be simplified by applying the double- and half-angle formulae.
  • The double-angle formulae are a special case of the sum formulae, where $\alpha = \beta$.
  • The double-angle formula for cosine can be derived similarly, and is:
  • We will apply the double-angle formula for sine: $\sin(2\theta) = 2\sin \theta \cos \theta $.
  • The half-angle formulae can be derived from the double-angle formulae.

• Experimental Probabilities

  • If a sample of $x$ trials is observed that results in an event, $e$, occurring $n$ times, the probability of event $e$ is calculated by the ratio of $n$ to $x$.
  • For example, if we flip a coin $10$ times, we might expect it to land on heads $5$ times, or half of the time.
  • Rolling a six-sided die one hundred times it's entirely possible that well over $\frac{1}{6}$ of the rolls will land on $4$.
  • If $1000$ draws are taken and the first number drawn is $5$, there are $999$ draws left to draw a $5$ again and thus have experimental data that shows double the expected likelihood of drawing a $5$.

• Direct Variation

  • For example, doubling y would result in the doubling of x.

• Introduction to Hyperbolas

  • The intersection of a right circular double cone with a plane perpendicular to the base of the cone
  • We need to square both sides of this equation multiple times if we want the variables to escape their square roots.

• Combined Variation

  • If x and y are in direct variation, and x is doubled, then y would also be doubled.
  • For example, if x and y are inversely proportional, if x is doubled, then y is halved.

• Common Bases of Logarithms

  • Logarithms were originally invented by John Napier (1515-1617) to aid in arithmetical computations at a time when modern day calculators were not in use.
  • Binary logarithms are useful in any application that involves the doubling of a quantity, and particularly in computer science with the use of integral parts.