Derivation of Formula | MATHalino reviewer about Derivation of Formula

The product of the first $n$ terms of a Geometric Progression is given by the following:

Given the first term $a_1$ and last term $a_n$:

$P_n = \sqrt{(a_1 \times a_n)^n}$

Given the first term $a_1$ and the common ratio $r$

$P_n = {a_1}^n \times r^{n(n - 1)/2}$

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Symbols and Notations
γ, γ_m = Unit weight, bulk unit weight, moist unit weight
γ_d = Dry unit weight
γ_sat = Saturated unit weight
γ_b, γ' = Buoyant unit weight or effective unit weight
γ_s = Unit weight of solids
γ_w = Unit weight of water (equal to 9810 N/m³)
W = Total weight of soil
W_s = Weight of solid particles
W_w = Weight of water
V = Volume of soil
V_s = Volume of solid particles
V_v = Volume of voids
V_w = Volume of water
S = Degree of saturation
w = Water content or moisture content
G = Specific gravity of solid particles

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The length of arc in polar plane is given by the formula:

$\displaystyle s = \int_{\theta_1}^{\theta_2} \sqrt{r^2 + \left( \dfrac{dr}{d\theta} \right)^2} ~ d\theta$

The formula above is derived in two ways.

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The length of arc in rectangular coordinates is given by the following formulas:

$\displaystyle s = \int_{x_1}^{x_2} \sqrt{1 + \left( \dfrac{dy}{dx} \right)^2} \, dx$ and $\displaystyle s = \int_{y_1}^{y_2} \sqrt{1 + \left(\dfrac{dx}{dy} \right)^2} \, dy$

See the derivations here: http://www.mathalino.com/reviewer/integral-calculus/length-arc-xy-plane…

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The three-moment equation gives us the relation between the moments between any three points in a beam and their relative vertical distances or deviations. This method is widely used in finding the reactions in a continuous beam.

Consider three points on the beam loaded as shown.

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Quadrilateral circumscribing a circle (also called tangential quadrilateral) is a quadrangle whose sides are tangent to a circle inside it.

Area,

$A = rs$

Where r = radius of inscribed circle and s = semi-perimeter = (a + b + c + d)/2

Derivation for area

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For a triangle of given three sides, say a, b, and c, the formula for the area is given by

$A = \sqrt{s(s - a)(s - b)(s - c)}$

where s is the semi perimeter equal to P/2 = (a + b + c)/2.

Read more about Derivation of Heron's / Hero's Formula for Area of Triangle
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Ptolemy's theorem for cyclic quadrilateral states that the product of the diagonals is equal to the sum of the products of opposite sides. From the figure below, Ptolemy's theorem can be written as

$d_1 d_2 = ac + bd$

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For a cyclic quadrilateral with given sides a, b, c, and d, the formula for the area is given by

$A = \sqrt{(s - a)(s - b)(s - c)(s - d)}$

Where s = (a + b + c + d)/2 known as the semi-perimeter.

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The lateral area of frustum of a right circular cone is given by the formula

$A = \pi (R + r) L$

where
R = radius of the lower base
r = radius of the upper base
L = length of lateral side

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