PLUMBING ARITHMETIC

Time (minutes)

- you can use this formula for clock problems, when you are asked what time will the clock of hands will have a certain angle etc etc.

2/11 x ( θ new +/- θ reference)

- use this when the question is asking how many minutes after a certain time will the clock angle be _____.

Clock Angle = 11M - 60H/2

- use this to determine the angle between the hour and minute hand of a clock.

arithmetic progression(sequence)

Sequence of numbers in which the difference of any two adjacent terms is constant

Arithmetic progression formula

For the nth term of Arithmetic Progression

An = A1 + (n - 1) d

where;

An = nth term

A1 = 1st term

D = common difference

Sum of terms of an Arithmetic Progression

S = n/2 (A1 + An)

where;

A1 = first term

An = last term

n = number of terms in the sequence

Geometric Progression Definition

Sequence of numbers where any two adjacent terms have the same ratio

nth term of a geometric sequence

An = A1 x r^(n-1)

where;

An = nth term

A1 = first term

r = common ratio

n = number of terms

Sum of terms in a Geometric sequence

Sum = A1 + (1 - r^n/1 - r )

Permutation

act of arranging objects or numbers in order.

Permutation Formula

nPr = n!/(n-r)!

Combination

way of selecting objects or numbers from a group of objects or collection. Order of objects does not matter

Combination Formula

nCr = n!/(n-r)!r!

Mixture

C1V1 + C2V2 = C3V3

where:

C1 and C2 = starting concentration

V1 and V2 = starting volume

C3 = Final Concentration

V3 = Final Volume

Work Rate (Formula)

Rate of Work = 1/Time Taken

1/A's time + 1/B's time = 1/Total Time Taken

Other useful formulae:

Man Power(Man Days/Output)

Trigonometric Functions

Sin θ = Opposite/Hypotenuse

Cos θ = Adjacent/Hypotenuse

Tan θ = Opposite/Adjacent

Pythagorean Identities

sin^2 θ + cos^2 θ = 1

1 + cot^2 θ = csc^2 θ

1 + tan^2 θ = sec^2 θ

Quotient Identities

tan θ = sin θ / cos θ

cot θ = cos θ / sin θ

SIN θ FUNCTIONS

sin θ = opposite/hypotenuse

sin θ = 1 / csc θ

sin θ = cos (90° - θ)

d/dx sin (x) = cos (x)

COS θ FUNCTIONS

cos θ = adjacent/hypotenuse

cos θ = 1/sec θ

cos θ = sin (90° - θ)

d/dx cos (x) = - sin (x)

CSC θ FUNCTIONS

csc θ = 1/sin θ

csc θ = sec (90° - θ)

d/dx csc (x) = - csc (x) cot (x)

TAN θ FUNCTIONS

tan θ = opposite/adjacent

tan θ = 1/cot θ

tan θ = cot (90° - θ)

d/dx tan (x) = sec^2 (x)

SEC θ FUNCTIONS

sec θ = 1/cos θ

sec θ = csc (90° - θ)

d/dx sec(x) = sec (x) tan (x)

COT θ FUNCTIONS

cot θ = 1/tan θ

cot θ = tan (90° - θ)

d/dx cot(x) = - csc ^2 (x)

Degrees

- Most common unit of angle

Mil

- short for milliradian

- old military measurement of angle, used for artillery calculations

Grad

- alternative unit for degree not commonly used

Radian

angle measurement equal to the length of an arc divided by the radius of the circle of arc

Degrees to Radians

Radians = Π/180 x (Degrees)

Radians to Degrees

Degrees = 180/Π x (Radians)

Number of Times the hands of the clock are at 90 degrees in a day

44

Number of times the hands of the clock overlap in a day

22

Number of times the hands of the clock are in opposite in a day

22

Equilateral

- 3 equal angles

- 3 equal sides

Scalene

- no equal sides

- no equal angles

Isosceles

- 2 equal sides

- 2 equal angles

Right Triangle

- 1 right angle

Pythagorean Theorem

c² = a² + b²

Law of Cosines

a² = b² + c² - 2bc (cos A)

b² = a² + c² - 2ac (cos B)

c² = a² + b² - 2ab (cos C)

Note:

A, B, C = are segment measurements/ sides

Law of Sines

sin A/a = sin B/b = sin C/c

Area of a Triangle

Height and Base Given:

= 1/2 x bh

3 Sides Given

√s(s-a)(s-b)(s-c)

where "s" is the semi=perimeter of the triangle

a,b,c are the sides of the triangle

Heron's Formula

s = a + b + c/2

Chord

Segment connecting 2 points in a circle

Secant

A line intersecting a circle TWICE

Tangent

A line intersecting a circle ONCE

Area of a Circle

πr²

Circumference of a Circle

If Radius is given:

2πr

if diameters is give:

πd

Area of a Sector

θ /360 π r ²

Arc Length

θ x radius

Ellipse

Created by a plane intersecting a cone at an angle of its base.

ALL Ellipses have TWO FOCI or focal points.

Area of an Ellipse

πab

where:

a = horizontal axis

b = vertical axis

Perimeter of the ellipse

2π (√(a² + b²)/2)

Quadrilateral

a polygon with

4 SIDES

4 ANGLES

4 VERTICES

Parallelogram

Opposite sides are parallel and equal

Area = b x h

Square

All angles at 90°

All sides are equal

Area = s x s

Rhombus

All sides equal, opposite sides are parallel

A = ½ (d₁x d₂)

Rectangle

All angles at 90°

Opposite sides are equal

Area = l x w

Trapezium

One pair of parallel sides

Area = A = ½ (a x b) x h

Kite

2 equal angles

two pairs of adjacent equal-length edges

Area = ½ (d₁x d₂)

POLYGON

a plane figure composed of finite line segments connected to each other to form a polygonal chain

Polygon Nomenclature

1 - hena

2 - di

3 - tri

4 - tetra

5 - penta

6 - hexa or exa

7 - hepta

8 - octa

9 - nona or ennea

10 - deca

11 - undeca or hendeca

12 - dodeca

13 - triskaideca

14 - tetradeca

15 - pentadeca

16 - hexadeca

17 - heptadeca

18 - octadeca

19 - enneadeca

20 - icosa

Polygon Nomenclature - Tens

20 - icosa

30 - triconta

40 - tetraconta

50 - pentaconta

60 - hexaconta

70 - heptaconta

80 - octaconta

90 - enneaconta

100 - hecta

Polygon Nomenclature Format

Prefix for tens + kai - prefix for units + gon

Example:

83 - Octacontakaitrigon

257 - dihectapentacontakaiheptagon

Polygon Area

Area if Apothem and Perimeter is given:

Area = ½ x apothem x perimeter

if side is given:

Area = ns²/ 4tan (180/n)

Polygon (Sum of Interior Angles)

Sum of Interior Angles:

(n - 2)(180°)

Interior Angle

((n - 2)(180°))/2

Polygon (Sum of Exterior Angles)

360°

To get the Angle measurement of each angle of a polygon:

360° / no. of sides

Polygons (Number of Diagonals)

Number of Diagonals in Polygons:

n(n-3)/2

Two Polygons are similar if:

a. all pairs of corresponding angles are congruent

b. all pairs of corresponding sides are parallel

Polyhedron

- three dimensional figure composed of flat polygonal faces, edges, and vertices

Platonic solids

- any of the five geometric solids composed of identical polygonal faces, regular polygons meeting at the same three dimensional angles.

The 5 Platonic Solids

THODI

Tetrahedron

Hexahedron

Octahedron

Dodecahedron

Icosahedron

Tetrahedron

4 Faces

6 Edges

4 Vertices

Face Shape = Triangle

Hexahedron or Cube

6 Faces

12 Edges

8 Vertices

Face Shape: Square

Octahedron

8 Faces

12 Edges

6 Vertices

Face Shape: Triangle

Dodecahedron

12 Faces

30 Edges

20 Vertices

Face Shape: Pentagon

Icosahedron

20 Faces

30 Edges

12 Vertices

Face Shape: Triangle

FRUSTUM

- a solid object formed when a plane cuts through at the apex of a cone or pyramid. The plane is parallel to the base of the cone of pyramid

Surface Area

Surface is area is equal to the sum of the base of the solid and the lateral areas.

Volume of a Frustum of a Cone

V = ⅓ π H(R₁² + R₂² + R₁² R₂²)

where:

H - height of the cone

R₁ = Radius of the Larger Base

R₂ = Radius of the Smaller Base

V = Volume

Surface Area of a Frustum of a Cone

Surface Area = Lateral Area + Area of the Bases

where:

Area of the bases = πR₁² + πR₂²

Lateral Area = ½ (C₁ + C₂) L

where:

C₁ = Circumference of the Larger base

C₂ = Circumference of the Smaller base

L = Slanted height

or

Lateral Area = π (R₁ + R₂) L

where

R₁ = Radius of the Larger base

R₂ = Radius of the Smaller base

L = Slanted height

Note:

L can be given in the problem, but you can also solve using the Pythagorean Theorem.

Volume of the Frustum of a Pyramid

V = ⅓ H(A₁ + A₂ + √A₁ A₂)

where

A₁ = Area of the larger base

A₂ = Area of the smaller base

H = Height

Surface Area of a Frustum of a Pyramid

Surface Area = Area of the bases + Lateral Area

where

Area of the Bases = A₁ + A₂

Lateral Area = ½ (P₁ + P₂) L

where:

P₁ = Perimeter of the Larger base

P₂ = Perimeter of the Smaller base

L = Slanted height

or

Lateral Area = ½n (B₁ + B₂) L

where

n = number of lower base edges

B₁ = Upper Base Edge

B₂ = Lower Base Edge

L = Slanted height

Parallelipiped (Box)

Volume = l x w x h

Total Surface Area = 2 lw + 2 lh + 2wh

Sphere

Volume = 4/3 π R³

Surface Area = 4 π R²

Pyramid

Volume = ⅓ (Area of the base) x H

Area of the Base = s x s or l x w

Surface Area = Area of the Base + Lateral Area

where

Lateral Area = ½ (B) x L

where

B = side of the base touched by the triangle

L = slanted Height (if not given, can be solved using Pythagorean Theorem)

Cylinder

Volume = π R² H

Total Surface Area = Area of the Bases + Lateral Area

Total Surface Area = 2 (π R²) + (2πR x H)

Cone

Volume = ⅓π R² H

Surface Area = Area of the Base + Lateral Area

Surface Area = π R² + π RL

where

L = slanted height ( if not given, can be solved using Pythagorean Theorem)

LINEAR EQUATIONS

- an algebraic equation where each term has an exponent of 1

- when graphed, always results in a straight Line

Standard Form of a Linear Equation

Ax + By = C

Slope Intercept Form of a Linear Equation

y = mx + b

where m = slope

b = y-intercept (where the graph intersects y axis)

Slope of a Line (Formula)

It is the "m" in the equation, y = mx + b

m = rise/run

m = change in y/change in x

m = y₂-y₁/x₂-x₁

Distance between a line and a point (Linear Equation)

Dline&point = |A(x) + B(y) + C| / √(A²-B²)

Distance between two points (Linear Equation)

Dpoints = √(change in y)² + (change in x)²

Dpoints = √(y₂-y₁)² + (x₂-x₁)²

QUADRATIC EQUATION

- an equation containing a single variable of a second degree (exponent of 2)

Standard Form of a Quadratic Form

ax² + bx = c = 0

Quadratic Formula

- b +/- √((b²) - 4ac)/2a

Circle

set points where all of their distance from a single point is constant.

Parabola

Locus of point which moves that is always equidistant from a fixed point (foci) and a from a fixed line (directrix or axis)

Ellipse

Locus of point which moves so that the sum of the distances from the two foci is always constant is equal to the length of the major axis.

Hyperbola

locus of point in a place so that the difference of its distance from two fixed points is (foci) is constant.

Physics

Study of matter, motion, energy and force.

Scalar Quantity

Only has magnitude

Vector Quantity

Has both magnitude and direction

Scalar Quantities (Examples)

Length, Area, Volume

Speed

Mass, Density

Pressure

Temperature

Energy, Entropy

Work, Power