proportional

Examples of proportional in the following topics:

• Direct and Inverse Variation

  • The two variables may be considered directly proportional.
  • In fact, two variables are said to be inversely proportional when an operation of change is performed on one variable and the opposite happens to the other.
  • 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.
  • An inversely proportional relationship between two variables is represented graphically by a hyperbola.

• Direct Variation

  • When two variables change proportionally, or are directly proportional, to each other, they are said to be in direct variation.
  • This can also be called directly proportional.

• Graphs of Exponential Functions, Base e

  • The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change (i.e., percentage increase or decrease) in the dependent variable.
  • For example, because a radioactive substance decays at a rate proportional to the amount of the substance present, the amount of the substance present at a given time can be modeled with an exponential function.
  • Also, because the the growth rate of a population of bacteria in a petri dish is proportional to its size, the number of bacteria in the dish at a given time can be modeled by an exponential function such as $y=Ae^{kt}$ where $A$ is the number of bacteria present initially (at time $t=0$) and $k$ is a constant called the growth constant.

• Solving Equations with Rational Expressions; Problems Involving Proportions

  • Rational expressions, like proportions, are extremely useful applications of algebra, that can be solved using simple algebraic techniques.
  • Solve equations with rational expressions (proportions) by finding the LCD or by cross-multipliation

• Combined Variation

  • The two variables may be considered directly proportional.
  • Two variables are said to be in inverse variation, or are inversely proportional, when an operation of change is performed on one variable and the opposite happens to the other.
  • For example, if x and y are inversely proportional, if x is doubled, then y is halved.

• Exponential Decay

  • Exponential decay is the result of a function that decreases in proportion to its current value.
  • Consider the decrease of a population that occurs at a rate proportional to its value.

• Basics of Graphing Exponential Functions

  • The exponential function $y=b^x$ where $b>0$ is a function that will remain proportional to its original value when it grows or decays.
  • When $b>1$ the function grows in a manner that is proportional to its original value.
  • When $0>b>1$ the function decays in a manner that is proportional to its original value.

• Applications of the Parabola

  • What Galileo discovered and tested was that when gravity is the only force acting on an object, the distance it falls is directly proportional to the time squared.
  • This is because air resistance is a force acting on the object, and is proportional to the object's area, density, and speed squared.

• The Law of Sines

  • We can use the following proportion from the law of sines:
  • Set up another proportion:

• Experimental Probabilities

  • An alternative estimate could be found by multiplying the proportion of men who are over six feet in height with the proportion of men who prefer strawberry jam to raspberry jam, but this estimate relies on the assumption that the two conditions are statistically independent.