MATH-compressed.pdf

# Lesson 2.5 The Set of Real Numbers - **When 2 is used as an exponent, the base is squared.** - **When 3 is used as an exponent, the base is cubed.** - **A square root of a given number is a number which square is the given number.** - In Symbols, if $a^2 = b$ the number $a$ is called a squ...

# Lesson 2.5 The Set of Real Numbers - **When 2 is used as an exponent, the base is squared.** - **When 3 is used as an exponent, the base is cubed.** - **A square root of a given number is a number which square is the given number.** - In Symbols, if $a^2 = b$ the number $a$ is called a square root of $b$. - **The divide-and-average method can be used to approximate square roots.** - **An irrational number is a number that cannot be expressed in the form $\frac{a}{b}$, where $a$ and $b$ are integers, and $b$ is not equal to 0.** - **Natural Numbers (N)** are numbers used for counting. - **Whole Numbers (W)** are numbers formed by adding 0 to the set of natural numbers. - **Integers (Z)** are formed by adding the negatives of the natural numbers to the set of whole numbers. - **The set of Rational Numbers (Q)** is the set of all members, which can be expressed in the form $\frac{a}{b}$, where $a$ and $b$ are integers, $b ≠ 0$. The decimal representation of a rational number either terminates or repeats. ## Lesson 2.1 Irrational Numbers - **Algebra** is defined as a branch of mathematics that generalizes the facts in arithmetic. - In Spoken language, letters and punctuations are used to create words and phrases. In the language of Algebra, letters along with numbers and operation symbols are used to create expressions. - **Algebraic Term** - either a single number, a letter, or a product of several numbers or letters. - **Algebraic Expression** - a statement containing one or more terms connected by plus or minus signs. - **Factors** - the numbers and symbols in a product. - **Literal Factor** - a letter used as a factor. - **Numerical Coefficient** - the number in an algebraic term. - **Literal Coefficient** - a letter used to represent a number. - **Exponent** - a small number written to the right of and slightly above another number or letter to indicate how many times the latter is used as a factor. - **Base** - a number or letter which an exponent refers. - **Power** - the product of a number used two or more times as a factor. ## Lesson 2.2 Measuring Lengths, Area, Surface Area, Capacity and Mass - 12 inches = 1 foot - 3 feet = 1 yard - 5 feet = 1 pace - 264 paces = 1 furlong - 8 furlongs = 1 mile - 24 furlongs = 1 league - A meter is about 39.37 inches long - a little longer than a yard. - The metric unit used for determining mass (weight) is called a gram. - The liter is the metric unit used for determining volume. - A kilogram is 1000 grams, whereas a hectoliter is 100 liters. A milliliter is $\frac{1}{1000}$ of a liter or 0.001 liter. An attometer is one quintillionth of a meter. - The basic unit of length for metric measure is the meter (m). Other commonly used units include the kilometer (km), the centimeter (cm), and the millimeter (mm). - The sum of the lengths of the sides of a polygon is called the perimeter (P) of the polygon. - A polygon is a closed figure which sides are called segments. A regular polygon is a polygon with all sides congruent and all angles congruent. - A circle is a set of all points in a plane that are at the same distance from a given point in the plane. The given point is called the center. - The circumference is the distance around a circle. - A face (F) is any flat surface. An edge (E) is a line segment where two faces meet. A vertex (V) is a point where several planes meet at a point. ## Lesson 2.3 Volume - **A prism** is a polyhedron with two identical parallel faces called bases. A prism has two bases, which are congruent polygonal regions lying in parallel planes. It can be named according to the shape of its bases. - **A right circular cone** is a solid that has many properties in common with a regular pyramid. - **The capacity of a container** is the amount of liquid it can hold. Volume and capacity are related in the metric system. A container that can hold one cubic centimeter of water has a capacity of one milliliter. - **In the metric system, the liter** is defined as the volume of a cubic decimeter. In other words, a liter is the capacity of a cube that is 1 decimeter long, 1 decimeter wide, and 1 decimeter high. The water in the liter container will fit exactly into the box with a 10 cm length, 10 cm width, and 10 cm height. One liter, 1 cubic decimeter, and 1000 cubic centimeters represent the same volume. - $1 L = 1 dm^3 = 1000 cm^3$ - **A cubic meter (m3)** is used to measure large volumes. Our water consumptions are measured in cubic meters. One cubic meter= 1.3 cubic yards. Extremely large quantities of liquids would be measured in terms of kiloliters. A kiloliter is equivalent to 1000 liters. - **Weight** is a measure of the earth's gravitational pull. **Mass** is the measure of the amount of matter that objects are made of. The mass of an object does not change, but its weight does. Weight and mass are not the same thing. However, on earth, the mass of an object is always proportional to its weight. - **The most common measures of weight** (mass) in the metric system are the kilogram, gram, and milligram. The gram is the basic unit of mass in the metric system. The mass (weight) of 1 milliliter of water is nearly equal to 1 gram, and a milliliter is the volume of a cube, which length, width, and height each measure 1 centimeter. - **A kilogram** is equivalent to 1000 grams. One pound is approximately 0.45 kilograms and 1 kilogram is approximately 2.2 pounds. Rice and other large foodstuff, such as fish and meat, are weighed in terms of kilograms. - $1 mg = \frac{1}{1000} gram$ - **A milligram** (mg) is a very small unit and is used only in medical prescriptions and some areas of science. - **For objects that are quite heavy, the metric tonne** unit is used. One metric tonne is equivalent 1000 kilograms. Since 1 kilogram is approximately 2.2 pounds, a metric tonne (1000 kilograms) is equivalent to 2 200 pounds, or 1.1 tons (ton = 2 000 pounds). ### Formulas - **Volume of a Cube:** $V = e^3$ - **Volume of a Rectangular Prism (or Cuboid):** $V = Bh$ - **Volume of a Cylinder:** Volume = $\pi r^2h$ cubic units. - **Volume of Pyramids:** $V = \frac{1}{3} \times Base area \times Height$ ## Properties of Real Numbers The properties of real numbers are: * **Closure Property of Addition:** The sum of any two real numbers is a real number. * **Closure Property of Multiplication:** The product of any two real numbers is a real number. * **Commutative Property of Addition:** Two real numbers can be added in any order. * **Associative Property of Addition:** When three real numbers are added, it makes no difference which two are added first. * **Associative Property of Multiplication:** When three real numbers are multiplied, it makes no difference which two are multiplied first. * **Distributive Property of Multiplication over Addition/Subtraction:** When Multiplication distributes over addition/subtraction. * **Identity Property of Addition:** Any number added to the identity element 0 will remain unchanged. 0 is the identity element for addition. * **Identity Property of Multiplication:** Any number multiplied to the identity element 1 will remain unchanged. 1 is the identity element for multiplication. * **Inverse Property of Addition:** The sum of a number, and its additive inverse (opposite) is the identity element 0. $a$ and (-$a$) are additive inverses. * **Inverse Property of Multiplication:** The product of a number, and its multiplicative inverse (reciprocal) is the identity element 1. $a$ and $\frac{1}{a}$ are multiplicative inverses. **Example 10** Show how a change of order (Commutative Property) makes it possible to do problems mentally: - $35+84+65 = (35+65) + 84 = 100 + 84 = 184$ - $4 \times 23 \times 25 = (4 \times 25) \times 23 = 100 \times 23 = 2300$ - $40 + 49 + 60 = (40 +60) + 49 = 100 + 49 = 149$ - $6 \times 12 \times 5 = (6 \times 5) \times 12 = 30 \times 12 = 360$ **Example 11** Rewrite the numbers so that each multiplication can be done mentally: - $5(6.04) = 5(6 + 0.04) = 30 + 0.20 = 30.20$ - $9(12.06) = 9(12 + 0.06) = 108 + 0.54 = 108.54$ ## The Set of Real Numbers **Example 3** Consider the following set of numbers: {-9, 1, 0, 0.3, √7, π, 8.4, √100} List all the numbers in the set that are: - **Natural Numbers:** √100 - **Whole Numbers:** √100, 0 - **Integers:** √100, 0, -9 - **Rational Numbers:** √100, 0, -9, 0.3, 8.4 - **Irrational Numbers:** √7, π - **Real Numbers:** All of the numbers. **Example 4** Check the set(s) to which each number belongs: | Set | -12 | √21 | -3 | 5 | 3.45 | 7 | |:---|:---:|:---:|:---:|:---:|:---:|:---:| | Natural Numbers | | | | ✓ | | ✓ | | Whole Numbers | | | | ✓ | | ✓ | | Integers | ✓ | | ✓ | ✓ | | ✓ | | Rational Numbers | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | | Irrational Numbers | | ✓ | | | | | | Real Numbers | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | **Example 5** On a number line, approximate the location of √2, -2√3, and π. - √2 ≈ 1.414 - -2√3 ≈ -3.46 - π ≈ 3.14 **Example 6** Oil flows through a pipe at a rate of 3 cubic feet per second. If the pipe has an internal diameter of 6 inches, calculate the volume of the oil flowing out per minute. - The diameter is 6 inches, so the radius (r) is 3 inches, which is $3 in \times \frac{1 ft}{12 in} = 0.25 ft$ - $A = \pi r^2 \approx 3.14(0.25)^2 ft^2 = 0.19625 ft^2$ - The volume of oil flowing per second is $3 ft^3/sec$ - The volume of oil flow per minute is $ 3 \times 60 = 180 ft^3/min$ **Example 7** A square pyramid has a base area of 25 cm² and a height of 9 cm. Find the volume of the pyramid. - $Volume of Pyramid = \frac{1}{3} \times Base \times height = \frac{1}{3} \times 25 cm^2 \times 9 cm = 75cm^3$ **Example 8** The volume of a rectangular prism is 560 cm³. A square pyramid that has a base area of 25 cm² and a height of 6 cm is placed inside it. Determine the volume of the space outside the pyramid. - $Volume of Pyramid = \frac{1}{3} \times Base area \times height = \frac{1}{3} \times 25 cm^2 \times 6 cm = 50 cm^3$ - $Volume outside of the pyramid = Volume of the rectangular prism - Volume of the pyramid = 560 cm^3 - 50 cm^2 = 510 cm^3$ ## The Set of Real Numbers **Example 1** The set of **Irrational Numbers** (Q') is the set of numbers which decimal representations are neither terminating nor repeating. These numbers cannot be expressed as a quotient of integers. **Example 2** Find all the subsets of the set $X = { 1, 2, 3}$. - Form all the subsets with no elements: Ø. - Form all the subsets with one element: {1}, {2}, {3}. - Form all the subsets with two elements: {1, 2}, {1, 3}, {2, 3}. - Form all the subsets with three elements: {1, 2, 3}. Set X has eight subsets, namely, Ø, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}. Note: A set with 3 elements has 8 subsets ($2^3 = 8$). **Example 9** Given: U = {Arnold, Ben, Carlos, Dennis, Efren, Fred, Gerald, Henry, Isaac, Joseph} - A = {Arnold, Carlos, Dennis, Henry} - B = {Carlos, Dennis, Joseph} Use Venn Diagram to represent: - **A U B:** {Arnold, Carlos, Dennis, Henry, Joseph} - **A ∩ B:** {Carlos, Dennis} - **A - B:** {Arnold, Henry} - **A':** {Ben, Efren, Fred, Gerald, Isaac} - **B':** {Arnold, Ben, Efren, Fred, Gerald, Henry, Isaac} - **(A ∩ B)':** {Arnold, Ben, Efren, Fred, Gerald, Henry, Isaac} **Example 12** Complete the following: - $0.24 kL = 24L$. - $500 dL = 50 000 mL$ **Example 13** Convert: - $50 qt = 12.5 gal$ - $1.25 gal = 20c$ **Example 14** A person breathes 105 gallons of air in one hour. How many quarts of air does a person breathe in one minute? - There are 60 min in 1 hr. A person breathes $\frac{105}{60}$ or 1.75 gal of air in 1 min. - $1.75 gal \times \frac{4 qt}{1 gal} = 7 qt$. - A person breathes 7 qt of air in one minute. **Example 15** How many cups of orange juice do you need to prepare one gallon of fruit smoothie? - Amount of orange juice in 1 pint of smoothie: $1 pt = 2c = \frac{1}{4}c + \frac{1}{3}c + 1c = \frac{5}{12}c$ - Since there are 8 pt in a gallon, there are $8 \times \frac{5}{12} = \frac{10}{3}$ or 3 and $\frac{1}{3}$ cups of orange juice in 1 gal of fruit smoothie. **Example 16** Find the better estimate for the following: - The weight of a newborn infant (30 kg, 3 kg, or 300 kg) = 3 kg - A piece of paper (0.5 kg, 0.5 g, or 5 kg) = 0.5 g **Example 17** Convert: - $0.3 g = 300 mg$ - $125 cg = 0.125 dag$ ## The Volume of a Cube, Rectangular Prism, Cylinder, and Pyramids **Example 1** Express the following: - 110.5 cm³ to m³ = (110.5 x 0.000001) m³ = 0.0001105 m³ - 15 280 mm³ to cm³ = (15 280 x 0.001) cm³ = 15.28 cm³ - 4.2 m³ to cm³ = (4.2 x 1 000 000) cm³ = 4 200 000 cm³ **Example 2** Find the volume of the following figures: - **Volume of a Cube** $V = e^3 = 10^3 = 1000ft^3$ - **Volume of a Rectangular Prism:** V= Bh= (240)(9) = 2160 ft³ **Example 18** Convert: - 128 oz to lb = $128 oz \times \frac{1 lb}{16 oz} = 8 lb$ - 120 lb to oz = $120 lb \times \frac{16 oz}{1 lb} = 1920 oz$ - 25.5 oz to g = 25.5 x 28 = 714 g - 56 kg to lb = $56 kg \times 2.2 = 123.2 lb$ **Example 19** A model of a doghouse made up of a square pyramid on top of a prism is shown below. ABCD is a square with $AB = 8 cm$. The height of the prism is 12 cm and $EF = 17 cm$. Find the volume of the doghouse. - The height of the pyramid is $EF - CR = 17 cm - 12 cm = 5 cm$ - The volume of the pyramid is: $Volume of Pyramid =\frac{1}{3} \times Base area \times height= \frac{1}{3} \times 8^2 \times 5$ **Example 20** Given: A = {1, 2, 3, 4, 5, 6} B = {2, 4, 6}, and C = {1, 3, 5}. Find: - **A ∩ B:** {2, 4, 6} - **A ∩ C:** {1, 3, 5} - **B ∩ C:** {} - **(A ∩ B) ∩ C:** {} - **A ∩ (B ∩ C):** {} **Example 21** Given: U = {s, e, c, o, n, d, a, r, y}, A = {n, e, r, d}, B = {d, a, r, y}, and C = { } Find: - **A':** {s, c, o, a, y} - **B':** {s, e, o, n, c, i} - **C':** {s, e, c, o, n, d, a, r, y, } - **(A ∩ B)':** {s, e, o, c. i, y} - **(A U B)':** {i, o} - **A' ∩ C':** {} - **A' ∩ B':** {s, c, o, a, y} **Example 22** Given: A = {r, o, y, g, b, i, v} and B = {r, y, b}. Find: - **A - B:** {o, g, i, v} - **B - A:** {} **Example 23** Given: A = {a, e, i, o, u}, B = {a, b, c, d, e}, and C = {c, u, b}. Find: - **A U B:** {a, e, i, o, u, b, c, d} - **A U C:** {a, e, i, o, u, b, c} - **B U C:** { a, b, c, d, e, u} - **(A U B) U C:** { a, e, i, o, u, b, c, d} - **A U (B U C):** { a, e, i, o, u, b, c, d} **Example 24** Determine all the subsets of the set: - **{a, b}:** { }, {a}, {b}, { a, b } - **{1, 2, 3}:** { }, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3} **Example 25** Are sets A and B equal sets? - A = {t, e, n} - B = { n, e, t} Yes, A and B are equal. ## Measuring Perimeter and Circumference **Example 8** Find the perimeter of each polygon: - In a rectangle: $P=21+2w$ - In a regular octagon: $P = 8s$ ## Measurement **Example 1** Convert: - 15 hm to dm: 15 hm = 15,000 dm. - 543 m to km: 543 m = 0.543 km. **Example 4** Convert: - 8 ft to in = 8 ft $\times \frac{12 in}{1 ft} $= 96 in - 144 in to yd = $144 in \times \frac{1 yd}{36 in} = 4 yd$ **Example 7** How much is the lot for sale? - The lot has an area of 240 yd². - Convert 240 yd² to m² = $240 yd^2 \times \frac{1 m^2}{1.2 yd^2}$ = 200 m² - The price of the lot is: $200 m^2 \times P5 000.00/m^2$ = P1 000 000.00 **Example 11** Find the lateral area and the surface area of the given figures: - **Lateral Area of a prism** = $hp$ - **Surface Area of a prism** = $L + 2B$ ## Measuring Area and Surface Area **Example 1** Find the area of a rectangular region: - $A = lw$ ## Measuring Capacity **Example 5** A person breathes 105 gallons of air in one hour. How many quarts of air does a person breathe in one minute? - $1 gal = 4 qt$ - $105 gal \times \frac{4 qt}{1 gal}= 420 qt$ - $\frac{420 qt}{60 min}= 7 qt/min$ **Try it 4** Convert 7 yards to feet. $7 yd = 7 yd \times \frac{3 ft}{1 yd} = 21 ft $ ## Measuring Mass **Example 16** Find the better estimate for the following: - the weight of a newborn infant (30 kg, 3 kg, or 300 kg) - a piece of paper (0.5 kg, 0.5 g, or 5 kg) - The weight of a newborn infant is approximately 3 kg. - The weight of a piece of paper is approximately 0.5 g. **Example 17** Convert the following: - 0.3 g to mg (move 3 places to the right) = 300 mg - 125 cg to dag (move 3 places to the right) = 0.125 dag **Try it 12** - 235 L = 23.5 hL - 45 cl = 0.45 kL **Try it 13** - 5 qt = 10 pt - 10 pt = 80 fl. oz **Try it 14** A person breathes 105 gallons of air in one hour. How many quarts of air does a person breathe in 10 minutes? - $105 gal \times \frac{4 qt}{1 gal}= 420 qt$ - $\frac{420 qt}{60 min}= 7 qt/min$ - $7qt/min \times 10 minutes= 70 qt$ **Try it 15** How many cups of orange juice do you need to prepare two gallons of fruit smoothie? - Since 1 gal of fruit smoothie needs 3 and $\frac{1}{3}$ cups of orange juice, then 2 gal needs $2 \times 3\frac{1}{3} = 6\frac{2}{3}$ cups. **Try it 16** Find the better estimate for a loaf of bread (5 g, 500 g, or 5 kg). - The weight of a loaf of bread is approximately 500 g. **Try it 17** Convert the following: - 3.2 kg to dg (move 4 places to the right) = 32,000 dg - 2,400 mg to kg (move 3 places to the left) = 2.4 kg **Try it 1** Give the equivalent expression in meter, gram, or liter for each of the following: - centigram = 0.01 grams - deciliter = 0.1 liters **Try it 2 ** What prefix can be used to indicate each number? - 100 = hecto- - 10 = deka- ## The Volume of a Cube, Rectangular Prism, Cylinder, and Pyramids **Try it 1** Express the following: - 15 m³ to cm³ = 15 m³ x (100 cm/m )³ = 15 x 1,000,000 cm³ = 15,000,000 cm³ - 205 mm³ to dm³ = 205 mm³ x (0.1 cm/mm)³ x (0.1 dm/cm)³ = 0.000205 dm³ - 600 cm³ to m³ = 600 cm³ x (0.01 m/cm)³ = 0.0006 m³ **Try it 2** Find the volume of the following figures: - **Volume of a cylinder** = $\pi r^2h = 3.14 \times 10^2 \times 15 cm^3 = 4710 cm^3$ **Try it 3** A cylindrical container has a volume of 750 cu. cm. A rectangular pyramid with a base area of 20 sq. cm, and a height of 15 cm is placed inside it. Find the volume of the space outside the rectangular pyramid. - $Volume of the rectangular pyramid = \frac{1}{3} \times base area \times height = \frac{1}{3} \times 20 cm^2 \times 15 cm = 100 cm^3$ - $Volume of the space outside the rectangular pyramid = Volume of the cylindrical container - Volume of the rectangular pyramid = 750 cm^3 - 100 cm^3 = 650 cm^3$ **Try it 4** You have two cylinders. Cylinder 1 has a radius of 10 cm and a height of 30 cm. Cylinder 2 has a radius of 6 cm and a height of 40 cm. Find the volume of the space between the two cylinders when Cylinder 2 is placed inside Cylinder 1. - $Volume of Cylinder 1 = \pi r^2h = 3.14 \times 10^2 \times 30 cm^3 = 9420 cm^3$ - $Volume of Cylinder 2 = \pi r^2h = 3.14 \times 6^2 \times 40 cm^3 = 4521.6 cm^3$ - The volume of the space between the two cylinders is: $9420cm^3 - 4521.6 cm^3 = 4,898.4 cm^3$ **Try it 5** A cylindrical water tank has a radius of 15 inches and a height of 24 inches. A smaller cylindrical tank is placed inside it, with a radius of 8 inches and a height of 18 inches. Calculate the volume of the remaining empty space in the larger container. - $Volume of the larger cylindrical tank = \pi r^2h = 3.14 \times 15^2 \times 24 in^3 = 16,956 in^3$ - $Volume of the smaller cylindrical tank = \pi r^2h = 3.14 \times 8^2 \times 18 in^3 = 3,619.2 in^3$ - The volume of the empty space is: $16,956 in^3 - 3,619.2 in^3 = 13,336.8 in^3$ **Try it 6** Oil flows through a pipe at a rate of 8 liters per second. If the pipe has an internal diameter of 10 centimeters, find the volume of the oil flowing out per minute. - The diameter is 10 centimeters, so the radius is $10 cm \div 2 = 5 cm = 0.05 m$. - $A = \pi r^2 \approx 3.14 \times 0.05^2 m^2 \approx 0.00785m^2$ - Volume of oil flowing per second = $8L/s = 8 dm^3/s$ - Volume of oil flowing per minute = $480 dm^3/min$ **Try it 7 ** A rectangular pyramid has a base area of 30 cm² and a height of 7 cm. Find the volume of the pyramid. - $Volume of a pyramid = \frac{1}{3} \times base area \times height = \frac{1}{3} \times 30 cm^2 \times 7 cm = 70 cm^3$ **Try it 8** Find the volume of the following figures: - **Volume of a square pyramid** = $ \frac{1}{3} \times bh = \frac{1}{3} \times (3.5 cm)^2 \times 3.5 cm = 15.3125 cm^3 $ - **Volume of rectangular prism:** = lwh = $6 in \times 12 in \times 15 in = 1080 in^3$ **Example 1** What prefix can be used to indicate each number? * 1000: kilo - * 0.01: centi- **Example 2** Give the equivalent expression in meter, gram, or liter for each of the following: - millimeter = 0.001 meters - kilogram = 1000 grams **Try it 1** Give the equivalent expression in meter, gram, or liter for each of the following: - centigram = 0.01 grams - deciliter = 0.1 liters **Try it 2** What prefix can be used to indicate each number? - 100 = hecto- - 10 = deka- ## The Surface Area of Space Figures The total surface area of a cylinder is found by adding the areas of the circular bases of the cylinder and the lateral area of the cylinder. If you think of a cylinder as a can, the lateral area is the area of the label. When you lay the label flat, you see that it is rectangular in shape, with a length equal to the circumference of the can and a width equal to the height of the can. **The surface area of a space figure** is the sum of the areas of all the faces. ## Formulas for Surface Area - **Surface Area of a Cube** = $6e^2$ - **Surface Area of a Rectangular Solid** = $ 2lw + 2hw + 2hl $ - **Lateral Area of a Right Prism** = hp - **Surface Area of a Right Prism** = $ L + 2B$ - **Lateral Area of a Regular Pyramid** = $ \frac{1}{2}lp$ - **Surface Area of a Regular Pyramid** = $L + B$ **Example 11** Find the lateral area and the surface area of the given figures. - **Lateral Area of a right prism:** $hp = 16 cm \times 24 cm = 384cm^2$ - **Surface Area of a right prism:** $L + 2B = 384 cm^2 + 2(24 cm^2) = 432 cm^2$ - **Lateral Area of a right Rectangular Pyramid:** $\frac{1}{2}lp = \frac{1}{2}(40 cm) (12 cm) = 240 cm^2$ **Try it 11** Find the surface area of the given figures. For c and d, find also the lateral area: - **A cube:** $6s^2 = 6(8.5 yd)^2 = 433.5 yd^2$ - **A rectangular solid:** $2lw + 2wh + 2lh = 2(6 cm)(12 cm) + 2(12 cm)(4 cm) + 2 (4 cm)(6 cm) = 288 cm^2$ - **Lateral Area of a cone:** $\pi rl