derivation of formula

Derivation of Formula for Lateral Area of Frustum of a Right Circular Cone

005 Figure for the Derivation of Formula for Lateral Area of Right Circular Cone

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

 

AL = π (R + r)L

 

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

 

Derivation of Formula for Radius of Circumcircle

004 Figure for derivation of radius of circumcircle

The formula for the radius of the circle circumscribed about a triangle (circumcircle) is given by

R = \dfrac{abc}{4A_t}

where At is the area of the inscribed triangle.

 

Derivation of Formula for Radius of Incircle

003 Figure for derivation of radius of incircle

The radius of incircle is given by the formula

r = \dfrac{A_t}{s}

where At = area of the triangle and s = semi-perimeter.

 

Derivation of Cosine Law

002 Figure for the derivation of Cosine Law

The following are the formulas for cosine law for any triangles with sides a, b, c and angles A, B, C, respectively.

 

a^2 = b^2 + c^2 - 2bc\cos A
b^2 = a^2 + c^2 - 2ac\cos B
c^2 = a^2 + b^2 - 2ab\cos C

 

Derivation of Sine Law

001 First figure for the derivation of sine law

For any triangles with vertex angles and corresponding opposite sides are A, B, C and a, b, c, respectively, the sine law is given by the formula...

 

\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}

 

Derivation of Sum and Difference of Two Angles

Triangle used in sum and difference of two anglesThe sum and difference of two angles can be derived from the figure shown to the right.

 

Consider triangle AEF:
\cos \beta = \dfrac{\overline{AE}}{1}; \,\, \overline{AE} = \cos \beta

\sin \beta = \dfrac{\overline{EF}}{1}; \,\, \overline{EF} = \sin \beta

 

From triangle EDF:
\sin \alpha = \dfrac{\overline{DE}}{\overline{EF}}
\sin \alpha = \dfrac{\overline{DE}}{\sin \beta}
\overline{DE} = \sin \alpha \, \sin \beta

 

\cos \alpha = \dfrac{\overline{DF}}{\overline{EF}}
\cos \alpha = \dfrac{\overline{DF}}{\sin \beta}
\overline{DF} = \cos \alpha \, \sin \beta

 

Derivation of Formula for Sum of Years Digit Method (SYD)

The depreciation charge and the total depreciation at any time m using the sum-of-the-years-digit method is given by the following formulas:

Depreciation Charge:

d_m = (FC - SV) \dfrac{n - m + 1}{SYD}

Total depreciation at any time m

D_m = (FC - SV) \dfrac{m(2n - m + 1)}{2 \times SYD}

Where:
FC = first cost
SV = salvage value
n = economic life (in years)
m = any time before n (in years)
SYD = sum of years digit = 1 + 2 + ... + n = ½ n(1 + n)

 

Derivation of formula for volume of a frustum

Frustum of a pyramid and frustum of a cone

Frustum of a pyramid and frustum of a cone

 

The formula for frustum of a pyramid or frustum of a cone is given by

V = \dfrac{h}{3} \left[ \, A_1 + A_2 + \sqrt{A_1A_2} \, \right]

Where:
h = perpendicular distance between A1 and A2 (h is called the altitude of the frustum)
A1 = area of the lower base
A2 = area of the upper base
Note that A1 and A2 are parallel to each other.

Solution to Problem 341 | Torsion of thin-walled tube

Problem 341

Derive the torsion formula τ = Tρ / J for a solid circular section by assuming the section is composed of a series of concentric thin circular tubes. Assume that the shearing stress at any point is proportional to its radial distance.

 

Derivation of the Double Angle Formulas

The Double Angle Formulas can be derived from Sum of Two Angles listed below:
\sin (A + B) = \sin A \, \cos B + \cos A \, \sin B \, \to \, Equation (1)
\cos (A + B) = \cos A \, \cos B - \sin A \, \sin B \, \to \, Equation (2)
\tan (A + B) = \dfrac{\tan A + \tan B}{1 - \tan A \, \tan B} \, \to \, Equation (3)

 

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