Examples of distance formula in the following topics:
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- The distance and the midpoint formulas give us the tools to find important information about two points.
- In analytic geometry, the distance between two points of the $xy$-plane can be found using the distance formula.
- The distance between points $(x_{1},y_{1})$ and $(x_{2},y_{2})$ is given by the formula:
- Substitute the values into the distance formula that is derived from the Pythagorean Theorem:
- The distance formula between two points, $(x_{1},y_{1})$ and $(x_{2},y_{2})$, shown as the hypotenuse of a right triangle
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- The equation for a circle is an extension of the distance formula.
- To find a formula for this, suppose that the center is the point $\left(a,b\right)$.
- According to the distance formula, the distance $c$ from the point $\left(a,b\right)$ to any other point $\left(x,y\right)$ is:
- This is the general formula for a circle with center $\left(a,b\right)$ and radius $r$.
- Notice that all we have done is slightly rearrange the distance formula equation.
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- There are formulae for the addition and subtraction of angles within each of the trigonometric functions.
- To see how these formulae are derived, we can place points on a diagram of a unit circle.
- The angles are equal, and so the distance between points $P$ and $Q$ is the same as between points $A$ and $B$.
- We can derive the following six formulae.
- Apply the formula $\cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta$:
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- The set of all points such that the ratio of the distance to a single focal point divided by the distance to a line (the directrix) is greater than one
- Then the difference of distances between $P$ and the two focal points is:
- where $a$ is the distance from the center (origin) to the vertices of the hyperbola.
- With this value for the difference of distances, we can choose any point $(x,y)$ on the hyperbola and construct an equation by use of the distance formula:
- The ellipse can be defined as all points that have a constant sum of distances to two focal points, and the hyperbola is defined as all points that have constant difference of distances to two focal points.
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- To find out how much it will cost you can use following formula:
- Plugging the known values into the above formula, we can determine that you will pay $500 in interest.
- There are many other common formulas that can be used for everyday computations.
- The formula relating gratuity (G), cost (c), and desired percent gratuity (r, expressed as a decimal).
- Use a given linear formula to solve for a missing variable
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- Through the work of scientists in the late 18th century, the main features of the electrostatic force—the existence of two types of charge, the observation that like charges repel, unlike charges attract, and the decrease of force with distance—were eventually refined, and expressed as a mathematical formula.
- The mathematical formula for the electrostatic force is called Coulomb's law after the French physicist Charles Coulomb (1736–1806), who performed experiments and first proposed a formula to calculate it.
- For example, it has been shown that the force is inversely proportional to distance between two objects squared (F∝1/r2) to an accuracy of 1 part in 1016.
- No exceptions have ever been found, even at the small distances within the atom.
- Generally, as the distance between ions increases, the energy of attraction approaches zero and ionic bonding is less favorable.
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- As the difference between the two prices or quantities increases, however, the accuracy of the formula decreases.
- The arc elasticity is obtained using this formula:
- The formula provided above would yield an elasticity of 0.4/(-1) = -0.4.
- It is the limit of the arc elasticity as the distance between the two points approaches zero, and hence is defined as a single point.
- The point elasticity can be calculated with the following formula:
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- An ellipse, which resembles an oval, is defined as all points whose distance from two foci add to a constant.
- The pen will touch every point on the cardboard such that the distance to one thumbtack, plus the distance to the other thumbtack, is exactly one string length.
- The cardboard is the "plane" in our definition, the thumbtacks are the "foci," and the string length is the "constant distance."
- The general formula for an ellipse is $\frac {x^2}{a^2}+\frac {y^2}{b^2} = 1$.
- Use formulas to determine the area and eccentricity of an ellipse
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- The set of all points in a plane that are the same distance from a given point forms a circle.
- The set of all points in a plane that are the same distance from a given point forms a circle.
- The point is known as the center of the circle, and the distance is known as the radius.
- You already know the formula for a line: y=mx+b.
- To understand the formula below, think of it as the y=mx+b of circles.
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- ., Einstein's famous formula $E=mc^2$), releasing this energy causes the mass of the nucleus to be lower than the total mass of the individual nucleons (leading to "mass deficit").
- The nuclear force is powerfully attractive between nucleons at distances of about 1 femtometer (fm) between their centers, but rapidly decreases to relative insignificance at distances beyond about 2.5 fm.
- At very short distances (less than 0.7 fm) it becomes repulsive; it is responsible for the physical size of nuclei since the nucleons can come no closer than the force allows.
- These nuclear forces are very weak compared to direct gluon forces ("color forces" or "strong forces") inside nucleons, and the nuclear forces extend over only a few nuclear diameters, falling exponentially with distance.
- Nevertheless, they are strong enough to bind neutrons and protons over short distances, as well as overcome the electrical repulsion between protons in the nucleus.