Geometric unsharpness formula

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Account the variation of gravity with latitude and elevation. In other words, geopotential height is a gravity-adjusted height. Variation in latitude is too small and usually is not taken in to account. The geopotential height is really a measure of the specific potential energy at a given geometric height relative to the Earth’s surface. It is used by meteorologists and in aviation. The relationship between the geopotential height H and the geometric height Z is given by the following formula (equation 18 in 1976 USSA), which is used in our calculations:For example, if Z = 86 km, which is the maximum geometric height in this calculator, the corresponding geopotential height will be H = 84.852 km. In this calculator, the geopotential height is calculated before pressure and temperature calculations.Speed of SoundThe speed of sound in air is about 343 m/s or 1.235 km/h or 767 mph. That means the sound can travel through the air one kilometer in about 3 seconds or a mile in about 5 seconds. The speed of sound in air depends mainly on its temperature; dependence on the sound frequency and air pressure is negligible. Water condensation at transonic speedThe speed of sound in dry air assuming it is an ideal gas at a relatively low pressure and density, which is correct for standard sea-level conditions, and also assuming that its temperature is lower or equal the room temperature is determined by the following formula, which is used in this calculator:where γ is the specific heat

A cone is a three dimensional geometric shape with one vertex and a circular base. The line form the centre of the base to the apex is the perpendicular height. Formula: Volume = (1/3) πr2h Table of Contents: Formula Formula Slant height of Cone (l) = Sqrt(r² + h²) Volume of Cone = (1/3)πr² h Curved Surface Area (CSA) of Cone = πrl Total Surface Area (TSA) of Cone = πr(l + r) where, r = radius, l = slant height, h = height, π = 3.14 A Cone is a geometric shape formed by having a circle at one end, usually at a base. It consists of all line segments joining to a single point to every point of a two-dimensional figure. Illustrated Examples: Find the volume, curved surface, and total surface area of a cone with the given radius 3 and height 4. Step 1: Find the slant height. Slant height (l) = Sqrt(r² + h²) = Sqrt(3² + 4²)= Sqrt(9 + 16) = Sqrt(25) = 5. Step 2: Find the volume. Volume = (1/3)πr² h = (1/3) * 3.14 * 3² * 4 = 0.33 * 113.04 = 37.68. Step 3: Find the curved surface area (CSA). CSA = πrl = 3.14 * 3 * 5 = 47.1. Step 4: Find the total surface area (TSA). TSA = πr(l + r) = 3.14 * 3(5 + 3) = 3.14 * 3(8) = 3.14 * 24 = 75.36. References: Kids Math: Finding the Volume and Surface Area of a Cone. What is Cone? - Definition, Facts & Example.

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Of the elements and its position/index. If we now sum all the elements of the sequence together, we get an infinite series:\[ S = 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \ldots \]Note that this particular series is known as the geometric series, where each consecutive term is related by a common ratio:\[ r = \frac{a_{n+1}}{a_n} \]Convergence and Divergence of SeriesAn infinite series can either converge (approach a definite, finite value) or diverge (approach an indefinite, infinite value). It may seem like an impossible problem, but we can perform several tests to determine whether a given series is convergent or divergent. The calculator uses the following:p-series TestRoot TestRatio TestIntegral TestLimit/Divergence TestIn some cases, some of the tests might be inconclusive. Further, some tests indicate convergence but do not provide the convergence value. There are also techniques specific to types of series, such as for a geometric series with common ratio r:\[ S_n = a + ar + ar^2 + \ldots + ar^{n-1} \]We have the formula for the sum up to of n terms of the series:\[ S_n = a \left ( \frac{1-r^{n+1}}{1-r} \right ) \, \, \text{where} \, \, r \neq 1 \]If r > 1, the infinite geometric series is divergent since the numerator $a(1-r^{n+1}) \to \infty$ as $n \to \infty$. However, if r \[ S = \frac{a}{1-r} \, \, \text{if} \, \, r Solved ExamplesExample 1Show that the harmonic series is divergent.\[ H = \left\{ a + \frac{1}{a+d} + \frac{1}{a+2d} + \frac{1}{a+3d} + \ldots \right\} \]SolutionThe summation form of the series at a, d=1 is:\[ H = \sum_{n \, = \, 1}^\infty \frac{1}{n} \]The limit test is inconclusive as $\lim_{n \to \infty} \frac{1}{n} = 0$ and it is only valid for limiting values greater than 0.The p-test states that for a sum of the form $\sum_{n \, = \, 1}^\infty \frac{1}{n^k}$, the series is divergent if $k \leq 1$ and convergent if k > 1. Here, the former is true so the series is divergent.The integral test further validates the p-series result:\[ \int_1^\infty \frac{1}{n} \cdot dn = \left. \ln n \right \rvert_1^\infty = \ln \infty \]So the series is divergent.Example 2Evaluate:\[ S = \sum_{n \, = \, 0}^\infty \frac{3^n+1}{4^n} \]SolutionLet $a_n = \frac{3^n+1}{4^n}$. Breaking it into two fractions:\[ a_n = \frac{3^n}{4^n} + \frac{1}{4^n} \]Then our sum is essentially the sum of two geometric series:\[ S = \underbrace{ \sum_{n \, = \, 0}^\infty \left ( \frac{3}{4} \right)^n }_\text{1$^\text{st}$ geometric series $G$} + \underbrace{ \sum_{n \, = \, 0}^\infty \left ( \frac{1}{4} \right)^n}_\text{2$^\text{nd}$ geometric series $G’$} \]Where $r = \frac{3}{4} = 0.75 \[ a = \left. \left( \frac{3}{4} \right)^n \right \rvert_{n \, = \, 0} = 1 \]\[ a’ = \left. \left( \frac{1}{4} \right)^n \right \rvert_{n \, = \, 0} = 1 \]Using the infinite geometric sum formula:\[ G = \frac{a}{1-r} = \frac{1}{0.25} = 4 \]\[ G’ = \frac{a’}{1-r’} = \frac{1}{0.75} = \frac{4}{3} \]\[ S = G + G’ = 4 + \frac{4}{3} = \frac{16}{3} \]So the series is convergent.Instantaneous Rate Of Change Calculator Math Calculators List >