Algebra, Area, and Volume

3-17 Conversion Large Numbers with Positive Powers of 10 Small Numbers with Negative Powers of 10 From standard notation to scientific notation From scientific notation to standard notation Move decimal place to the right Move decimal place to the left Move decimal place to the left Move decimal place to the right 1 + CT= 1 C1 1 C2 + 1 C3 CT = 1 = 1 = 1 10 + 66.66 + 20 1 C1 + 1 C2 + 1 C3 1 0.1 + 1 0.015 + 1 0.05 Therefore, CT = 1⁄96.66 = 0.01034 microfarad. The microfarad (10-6 farad) is a unit of measurement of capacitance. This is discussed in greater length in Chapter 12, Electricity. Use of Parentheses In algebraic equations, parentheses are used to group numbers or symbols together. The use of parentheses helps us to identify the order in which we should apply mathematical operations. The operations inside the parentheses are always performed first in algebraic equations.

Example: Solve the algebraic equation X = (4 + 3) 2. First, perform the operation inside the parentheses, which is, 4 + 3 = 7. Then complete the exponent calculation X = (7)2 = 7 × 7 = 49. When using more complex equations, which may combine several terms and use multiple operations, grouping the terms together helps organize the equation. Parentheses, ( ), are most commonly used in grouping, but you may also see brackets, [ ]. When a term or expression is inside one of these grouping symbols, it means that any operation indicated to be done on the group is done to the entire term or expression.

Example: Solve the equation N = 2 × [(9 ÷ 3) + (4 + 3) 2]. Start with the operations inside the parentheses ( ), then perform the operations inside the brackets [ ]. N = 2 × [(9 ÷ 3) + (4 + 3)2] N = 2 × [3 + (7)2] First, complete the operations inside the parentheses ( ). N = 2 × [3 + 49] N = 2 × [52] Second, complete the operations inside the brackets [ ]. N = 104 Order of Operation In algebra, rules have been set for the order in which operations are evaluated. These same universally accepted rules are also used when programming algebraic equations in calculators. When solving the following equation, the order of operation is given below: N = (62 – 54)2 + 62 – 4 + 3 × [8 + (10 ÷ 2)] + 25 + (42 × 2) ÷ 4 + 3⁄4 1. Parentheses. First you must do everything in parentheses, ( ), starting from the innermost parentheses. If the expression has a set of brackets, [ ], treat these exactly like parentheses. If you are working with a fraction, treat the top as if it was in parentheses and the denominator as if it were in parentheses, even if there is none shown.

From the equation above, completing the calculation in parentheses gives the following: N = (8)2 + 62 – 4 + 3 × [8 + (5)] + 25 + (84) ÷ 4 + 3⁄4, then N = (8)2 + 62 – 4 + 3 × [13] + 25 + 84 ÷ 4 + 3⁄4 2. Exponents. Next, clear any exponents. Treat any roots (square roots, cube roots, and so forth) as exponents. Completing the exponents and roots in the equation gives the following: N = 64 + 36 – 4 + 3 × 13 + 5 + 84 ÷ 4 + 3⁄4 3. Multiplication and Division. Evaluate all of the multiplications and divisions from left to right. Multiply and divide from left to right in one step. A common error is to use two steps for this (that is, to clear all of the multiplication signs and then clear all of the division signs), but that is not the correct method. Treat fractions as division. Completing the multiplication and division in the equation gives the following: N = 64 + 36 – 4 + 39 + 5 + 21 + 3⁄4 4. Addition and Subtraction. Evaluate the additions and subtractions from left to right. Like above, addition and subtraction are computed left to right in one step. Completing the addition and subtraction in the equation gives the following: X = 161 3⁄4 A commonly used acronym, PEMDAS, is used for remembering the order of operation in algebra. PEMDAS is an acronym for parentheses, exponents, multiplication, 3-18 division, addition, and subtraction. To remember it, many use the sentence, “Please Excuse My Dear Aunt Sally.” Always remember, however, to multiply/divide or add/subtract in one sweep from left to right, not separately.

Order of Operation for Algebraic Equations 1. Parentheses 2. Exponents 3. Multiplication 4. Division 5. Addition 6. Subtraction

Computing Area of Two-Dimensional Solids

Area is a measurement of the amount of surface of an object. Area is usually expressed in such units as square inches or square centimeters for small surfaces or in square feet or square meters for larger surfaces. of two-dimensional solids. Rectangle A rectangle is a four-sided figure with opposite sides of equal length and parallel to each other. [Figure 3-14] All of the angles are right angles. A right angle is a 90° angle. The rectangle is a very familiar shape in mechanics. The formula for the area of a rectangle is: area = length × width = l × w Example: An aircraft floor panel is in the form of a rectangle having a length of 24 inches and a width of 12 inches. What is the area of the panel expressed in square inches? First, determine the known values and substitute them in the formula.

a = l × w = 24 inches × 12 inches = 288 square inches Square A square is a four-sided figure with all sides of equal length and opposite sides are parallel to each other. [Figure 3-15] All angles are right angles. A right angle is a 90° angle. The formula for the area of a square is: area = length × width = l × w Since the length and the width of a square are the same value, the formula for the area of a square can also be written as: area = side × side = s2 Example: What is the area of a square access plate whose side measures 25 inches? First, determine the known value and substitute it in the formula.

a = l × w = 25 inches × 25 inches = 625 square inches Triangle A triangle is a three-sided figure. The sum of the three angles in a triangle is always equal to 180°. Triangles are often classified by their sides. An equilateral triangle has 3 sides of equal length. An isosceles triangle has 2 sides of equal length. A scalene triangle has three sides of differing lengths. Triangles can also be classified by their angles. An acute triangle has all three angles less than 90°. A right triangle has one right angle (a 90° angle). An obtuse triangle has one angle greater than 90°. Each of these types of triangles is shown in Figure 3-16.

The formula for the area of a triangle is area = 1⁄2 × (base × height) = 1⁄2 × (b × h) Example: Find the area of the obtuse triangle shown in area formula. a = 1⁄2 × (b × h) = 1⁄2 × (2'6" × 3'2") Next, convert all dimensions to inches: 2'6" = (2 × 12") + 6" = (24 + 6) = 30 inches 3'2" = (3 × 12") + 2" = (36 + 2) = 38 inches Now, solve the formula for the unknown value: a = 1⁄2 × (30 inches × 38 inches) = 570 square inches Parallelogram A parallelogram is a four-sided figure with two pairs of parallel sides. [Figure 3-18] Parallelograms do not necessarily have four right angles. The formula for the area of a parallelogram is: area = length × height = l × h Trapezoid A trapezoid is a four-sided figure with one pair of parallel sides. [Figure 3-19] The formula for the area of a trapezoid is: area = 1⁄2 (base1 + base2) × height 3-19 w = 12 l = 24 s = 25 Object Area Formula Figure Rectangle Square Parallelogram Trapezoid Circle Ellipse Wing area Triangle 3-14 3-15 3-18 3-19 3-20 3-21 3-22 3-16 3-17 a = l × w a = l × w or a = s2 a = l × h a = ½ (b1 + b2) × h a = π × r2 a = π × a × b a = s × c a = ½ (l × h) or a = ½ (b × h) or a = (b × h) ÷ 2 length × width length × width or side × side length × height ½ (base1 + base2) × height π × radius2 π × semi-axis A × semi-axis B span × mean chord ½ × (length × height) or ½ × (base × height) or (base × height) ÷ 2 Example: What is the area of a trapezoid in Figure 3-19 whose bases are 14 inches and 10 inches, and whose height (or altitude) is 6 inches? First, substitute the known values in the formula.

a = 1⁄2 (b1 + b2) × h = 1⁄2 (14 inches + 10 inches) × 6 inches a = 1⁄2 (24 inches) × 6 inches = 12 inches × 6 inches = 72 square inches Circle A circle is a closed, curved, plane figure. [Figure 3-20] Every point on the circle is an equal distance from the center of the circle. The diameter is the distance across the circle (through the center). The radius is the distance from the center to the edge of the circle. The diameter is always twice the length of the radius. The circumference, or distance around, a circle is equal to the diameter times π. circumference = c = d π The formula for the area of a circle is: area = π × radius2 = π × r2 Example: The bore, which is “inside diameter,” of a certain aircraft engine cylinder is 5 inches. Find the area of the cross section of the cylinder.

First, substitute the known values in the formula: a = π × r2 The diameter is 5 inches, so the radius is 2.5 inches. (diameter = radius × 2) a = 3.1416 × (2.5 inches) 2 = 3.1416 × 6.25 square inches = 19.635 square inches Ellipse An ellipse is a closed, curved, plane figure and is commonly called an oval. [Figure 3-21] In a radial engine, the articulating rods connect to the hub by pins, which travel in the pattern of an ellipse (i.e., an elliptical or orbital path). 3-20 Base = 3 ft 2 in Height = 2 ft 6 in b1 = 14" b2 = 10" Height = 6" Length Height EquilateralIsosceles Triangles Based on Sides Triangles Based on Angles Scalene Acute Right Obtuse Length of all sides are different Length of two sides are equal Length of all sides are equal Each angle is < 90° One angle is = 90° One angle is > 90° Wing Area To describe the shape of a wing [Figure 3-22], several terms are required. To calculate wing area, it is necessary to know the meaning of the terms “span” and “chord.” The wingspan, S, is the length of the wing from wingtip to wingtip. The chord is the average width of the wing from leading edge to trailing edge. If the wing is a tapered wing, the average width, known as the mean chord (C), must be known to find the area. The formula for calculating wing area is: area of a wing = span × mean chord Example: Find the area of a tapered wing whose span is 50 feet and whose mean chord is 6'8". First, substitute the known values in the formula.

a = s × c = 50 feet × 6 feet 8 inches (Note: 8 inches = 8⁄12 feet = 0.67 feet) = 50 feet × 6.67 feet = 333.5 square feet Units of Area A square foot measures 1 foot by 1 foot. It also measures 12 inches by 12 inches. Therefore, one square foot also equals 144 square inches (that is, 12 × 12 = 144). To convert square feet to square inches, multiply by 144. To convert square inches to square feet, divide by 144. A square yard measures 1 yard by 1 yard. It also measures 3 feet by 3 feet. Therefore, one square yard also equals 9 square feet (that is, 3 × 3 = 9). To convert square yards to square feet, multiply by 9. To convert square feet to square yards, divide by 9. Refer to Figure 3-23, Applied Mathematics Formula Sheet, for a comparison of different units of area.

Computing Volume of Three-Dimensional Solids Three-dimensional solids have length, width, and height. There are many three-dimensional solids, but the most common are rectangular solids, cubes, cylinders, spheres, and cones. V olume is the amount of space within a solid. V olume is expressed in cubic units. Cubic inches or cubic centimeters are used for small spaces and cubic feet or cubic meters for larger spaces. Rectangular Solid A rectangular solid is a three-dimensional solid with six rectangular-shaped sides. [Figure 3-24] The volume is the 3-21 Circumference Diameter (d) Radius (r) b a π = 3.1416 a = Length of one of the semi-axes b = Length of the other semi-axis Area = a = π x a x b Circumference = c = 2π a2 + b2 2 s c a = Wing area, ft.2 c = Average chord, ft.

s = Span, ft. number of cubic units within the rectangular solid. The formula for the volume of a rectangular solid is: volume = length × width × height = l × w × h In Figure 3-24, the rectangular solid is 3 feet by 2 feet by 2 feet. The volume of the solid in Figure 3-24 is = 3 ft × 2 ft × 2 ft = 12 cubic feet. Example: A rectangular baggage compartment measures 5 feet 6 inches in length, 3 feet 4 inches in width, and 2 feet 3 inches in height. How many cubic feet of baggage will it hold? First, substitute the known values into the formula. v = l × w × h = 5'6" × 3'4" × 2'3" = 5.5 ft × 3.33 ft × 2.25 ft = 41.25 cubic feet Cube A cube is a solid with six square sides. [Figure 3-25] A cube is just a special type of rectangular solid. It has the same formula for volume as does the rectangular solid, which is volume = length × width × height = L × W × H. Because all of the sides of a cube are equal, the volume formula for a cube can also be written as: volume = side × side × side = S3 Example: A large, cube-shaped carton contains a shipment of smaller boxes inside of it. Each of the smaller boxes is 1 ft × 1 ft × 1 ft. The measurement of the large carton is 3 ft × 3 ft × 3 ft. How many of the smaller boxes are in the large carton? First, substitute the known values into the formula.

v = l × w × h = 3 ft × 3 ft × 3 ft = 27 cubic feet of volume in the large carton Since each of the smaller boxes has a volume of 1 cubic foot, the large carton holds 27 boxes. Cylinder A solid having the shape of a can, a length of pipe, or a barrel is called a cylinder. [Figure 3-26] The ends of a cylinder are identical circles. The formula for the volume of a cylinder is: volume = π × radius2 × height of the cylinder = π r2 × h One of the most important applications of the volume of a cylinder is finding the piston displacement of a cylinder in a reciprocating engine. Piston displacement is the total volume (in cubic inches, cubic centimeters, or liters) swept by all of the pistons of a reciprocating engine as they move in one revolution of the crankshaft. The formula for piston displacement is given as: Piston Displacement = π × (bore divided by 2)2 × stroke × (# cylinders) 3-22 Length 2.54 centimeters 12 inches 3 feet 5,280 feet 0.0394 inches 0.62 miles 25.4 millimeters 30.48 centimeters 0.9144 meters 1,760 yards 1 inch 1 foot 1 yard 1 mile 1 millimeter 1 kilometer Weight 1 ounce 28.350 grams 1 pound 16 ounces 453.592 grams 0.4536 kilograms 1 ton 2,000 pounds 1 milligram 0.001 grams 1 kilogram 1,000 grams 2.2 pounds 1 gram 0.0353 ounces Area 6.45 square centimeters 144 square inches 9 square feet 43,560 square feet 640 acres 0.155 square inches 1.195 square yards 0.384 square miles 0.093 square meters 0.836 square meters 2.59 square kilometers 1 square inch 1 square foot 1 square yard 1 acre 1 square mile 1 square centimeter 1 square meter 1 square kilometer Volume 1 fluid ounce 29.57 cubic centimeters 1 cup 8 fluid ounces 1 pint 2 cups 16 fluid ounces 0.473 liters 1 quart 2 pints 4 cups 32 fluid ounces 0.9463 liters 1 gallon 4 quarts 8 pints 16 cups 128 ounces 3.785 liters 1 gallon 231 cubic inches 1 liter 0.264 gallons 1.057 quarts 1 cubic foot 1,728 cubic inches 7.5 gallons 1 cubic yard 27 cubic feet 1 board foot 1 inch x 12 inches x 12 inches Temperature °F to °C °C to °F Celsius = ⁵⁄9 × (°F − 32) Fahrenheit = ⁹⁄5 × (°C + 32) Conversion Factors The bore of an engine is the inside diameter of the cylinder.

The stroke of the engine is the length the piston travels inside the cylinder. [Figure 3-27] Example: Find the piston displacement of one cylinder in a multi-cylinder aircraft engine. The engine has a cylinder bore of 5.5 inches and a stroke of 5.4 inches. First, substitute the known values in the formula. v = π × r2 × h = (3.1416) × (5.5 ÷ 2)2 × (5.4) v = 23.758 × 5.4 = 128.29 cubic inches The piston displacement of one cylinder is 128.29 cubic inches. For an eight-cylinder engine, then the total engine displacement would be: Total displacement for 8 cylinders = 8 × 128.29 = 1026.32 cubic inches of displacement Sphere A solid having the shape of a ball is called a sphere.

[Figure 3-28] A sphere has a constant diameter. The radius (r) of a sphere is one-half of the diameter (d). The formula for the volume of a sphere is given as: 3-23 H L W L = 3 W= 2 H = 2 s Solid Volume Surface Area Figure 1-24 1-25 1-26 1-28 1-29 l × w × h s3 π × r2 × h /four.numerator⁄/three.denominator × π × r3 /one.numerator⁄/three.denominator × π × r2 × h 2 × [(w × l) + (w × h) + (l × h)] 6 × s2 2 × π × r2 + π × d × h 4 × π × r2 π × r × [r + (r2 + h2)½] Rectangle Solid Cube Cylinder Sphere Cone Formulas for Area of Two-Dimensional ObjectsOrder of Operation for Algebraic Equations Names and Symbols for Metric Prefixes Powers of Ten Prefix Means exa (10 18) peta (10 15) tera (10 12) giga (10 9) mega (10 6) kilo (10 3) hecto (10 2) deca (10 1) deci (10 −1) centi (10 −2) milli (10 −3) micro (10 −6) nano (10 −9) pico (10 −12) One quintillion times One quadrillion times One trillion times One billion times One million times One thousand times One hundred times Ten times One tenth of One hundredth of One thousandth of One millionth of One billionth of One trillionth of

Powers

of Ten Expansion Value Positive Exponents 106 105 104 103 102 101 100 1,000,000 100,000 10,000 1,000 100 10 1 10 x 10 x 10 x 10 x 10 x 10 10 x 10 x 10 x 10 x 10 10 x 10 x 10 x 10 10 x 10 x 10 10 x 10 10 Negative Exponents 10-1 10-2 10-3 10-4 10-5 10-6 1/10=0.1 1/100=0.01 1/1,000=0.001 1/10,000=0.0001 1/100,000=0.00001 1/1,000,000=0.000001 1/10 1/(10 x 10) 1/(10 x 10 x 10) 1/(10 x 10 x 10 x 10) 1/(10 x 10 x 10 x 10 x 10) 1/(10 x 10 x 10 x 10 x 10 x 10) 1. P arentheses 2. Exponents 3. Multiplication 4. Division 5. Addition 6. Subtraction c b a C B A c b a a2 + b2 = c2 Trigonometric Equations Pythagorean Theorem Use the acronym PEMDAS to remember the order of operation in algebra. PEMDAS is an acronym for parentheses, exponents, multiplication, division, addition, and subtraction.

To remember it, many use the sentence, “Please Excuse My Dear Aunt Sally.” Sine (sin) of angle A = opposite side (side a) hypotenuse (side c) Cosine (cos) of angle A = adjacent side (side b) hypotenuse (side c) Tangent (tan) of angle A = opposite side (side a) adjacent side (side b) v = 4⁄3 × π × radius3 = 4⁄3 × π × r3 or v = 1⁄6 × πd3 Example: A pressure tank inside the fuselage of a cargo aircraft is in the shape of a sphere with a diameter of 34 inches. What is the volume of the pressure tank? v = 4⁄3 × π × radius3 = 4⁄3 × (3.1416) × (34⁄2)3 = 1.33 × 3.1416 × 173 = 1.33 × 3.1416 × 4,913 v = 20,528.125 cubic inches Cone A solid with a circle as a base and with sides that gradually taper to a point is called a cone. [Figure 3-29] The formula for the volume of a cone is given as: v = 1⁄3 × π × radius2 × height = 1⁄3 × π × r2 × h Units of Volume Since all volumes are not measured in the same units, it is necessary to know all the common units of volume and how they are related to each other. For example, the mechanic may know the volume of a tank in cubic feet or cubic inches, but when the tank is full of gasoline, they are interested in how many gallons it contains. Refer to Figure 3-23, Applied Mathematics Formula Sheet, for a comparison of different units of volume.

Computing Surface Area of Three- Dimensional Solids The surface area of a three-dimensional solid is the sum of the areas of the faces of the solid. Surface area is a different