Geometry: Volume and Properties of a Cube
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Questions and Answers

What is the formula to calculate the volume of a cube?

  • Volume = edge × edge
  • Volume = edge × edge × edge (correct)
  • Volume = π × radius² × height
  • Volume = length × width × height

    • The volume of the sculpture is equal to 4913 in.³.

    True (A)

    What is the length of one edge of the cube if its volume is 4913 in.³?

    17 in.

    The volume of a cube is calculated by raising the length of one edge to the _____ power.

    <p>third</p> Signup and view all the answers

    Match the following terms with their definitions:

    <p>Volume = The amount of space occupied by an object Edge = A line segment where two faces meet Cube = A three-dimensional shape with equal sides Cube Root = A value that, when cubed, gives the original number</p> Signup and view all the answers

    What is the length of one side of a cube with a volume of 3375 cm³?

    <p>15 cm (D)</p> Signup and view all the answers

    The diagonal distance through the cube from one corner to the opposite corner is 19 cm.

    <p>False (B)</p> Signup and view all the answers

    What formula is used to calculate the diagonal distance through a cube?

    <p>d = s√3</p> Signup and view all the answers

    The formula for the volume of a cube is _____ cubic centimeters.

    <p>s^3</p> Signup and view all the answers

    Match the terms related to a cube:

    <p>Volume = Amount of space within a cube Diagonal = Distance between two opposite corners Side length = Length of one edge of the cube Surface area = Total area of all cube faces</p> Signup and view all the answers

    What is the area of a square with a side length of 5 cm?

    <p>25 cm² (C)</p> Signup and view all the answers

    A cube has vertices that are only in two dimensions.

    <p>False (B)</p> Signup and view all the answers

    What is the edge length of the cube mentioned?

    <p>5 cm</p> Signup and view all the answers

    The area of a square is calculated as the side length squared, so if the side length is ___ cm, the area is 25 cm².

    <p>5</p> Signup and view all the answers

    Match the following terms with the correct definitions:

    <p>Area = The measure of the surface within a shape. Edge = Line segment where two faces meet. Cube = A three-dimensional shape with six equal square faces. Square = A four-sided shape with equal sides and right angles.</p> Signup and view all the answers

    What effect does raising a number to the exponent -3 have?

    <p>It takes the reciprocal of the number cubed (B)</p> Signup and view all the answers

    The symbol for infinity is represented as '∞'.

    <p>True (A)</p> Signup and view all the answers

    What does multiplying the result by a number signify in relation to exponents?

    <p>Scaling the value</p> Signup and view all the answers

    To raise a number to the exponent of -3, you must first find the _____ of the number cubed.

    <p>reciprocal</p> Signup and view all the answers

    Match the mathematical operation with its description:

    <p>Exponentiation = Raising a number to a power Multiplication = Scaling a number by another number Reciprocal = The inverse of a number Infinity = A concept of an unbounded quantity</p> Signup and view all the answers

    What is the area of a square if the side length is 4 cm?

    <p>16 cm² (D)</p> Signup and view all the answers

    The area of a square increases quadratically with the increase in its side length.

    <p>True (A)</p> Signup and view all the answers

    What is the relationship between the side length of a square and its area?

    <p>The area of a square is equal to the side length squared.</p> Signup and view all the answers

    If the side length of a square is represented by 's', then the area is _____.

    <p>s²</p> Signup and view all the answers

    Match the following side lengths with their corresponding areas of a square:

    <p>2 cm = 4 cm² 3 cm = 9 cm² 5 cm = 25 cm² 6 cm = 36 cm²</p> Signup and view all the answers

    Which of the following numbers is both a perfect square and a perfect cube?

    <p>1 (A)</p> Signup and view all the answers

    The number 169 is a perfect square.

    <p>True (A)</p> Signup and view all the answers

    What is the cube root of 216?

    <p>6</p> Signup and view all the answers

    The number 1024 is a perfect _____ but not a perfect cube.

    <p>square</p> Signup and view all the answers

    Match the following numbers with their classification:

    <p>1 = Both perfect square and perfect cube 81 = Perfect square only 216 = Perfect cube only 1000 = Perfect cube only</p> Signup and view all the answers

    What percentage of its charge does a 12 V NiMH battery lose every month if not recharged?

    <p>30% (C)</p> Signup and view all the answers

    A 12 V nickel-metal hydride battery will not lose any charge if it remains uncharged for a month.

    <p>False (B)</p> Signup and view all the answers

    How often should a 12 V NiMH battery be recharged to avoid losing charge?

    <p>Every month</p> Signup and view all the answers

    If a 12 V NiMH battery is not recharged, it will lose approximately _____ of its charge in one month.

    <p>30%</p> Signup and view all the answers

    Match the following battery characteristics with their descriptions:

    <p>12 V NiMH = Commonly used in power tools Charge Loss = 30% per month if not recharged Recharge Frequency = Recommended every month Battery Type = Nickel-metal hydride</p> Signup and view all the answers

    What does the variable C represent in the formula C = (1/21)t?

    <p>Fraction of the surface area covered by algae (C)</p> Signup and view all the answers

    The formula C = (1/21)t indicates that algae coverage decreases over time.

    <p>False (B)</p> Signup and view all the answers

    If you want to find out when 25% of the pond was covered with algae, you need to calculate what value of t?

    <p>t = 21 * 0.25</p> Signup and view all the answers

    To find the time in weeks when the pond had 25% algae coverage, you would set C equal to _____ and solve for t.

    <p>0.25</p> Signup and view all the answers

    Match the following values with the correct interpretation in the context of the algae coverage formula:

    <p>C = 0.25 = 25% coverage of the pond t = 21 = Time measured in weeks C = 1 = 100% coverage of the pond C = 0 = No coverage of algae</p> Signup and view all the answers

    A cube has 12 edges.

    <p>True (A)</p> Signup and view all the answers

    What is the edge length of a cube if its volume is 125 cm³?

    <p>5 cm</p> Signup and view all the answers

    The volume of a cube is found by raising the length of one edge to the _____ power.

    <p>third</p> Signup and view all the answers

    What is the square root of 25?

    <p>5 (C)</p> Signup and view all the answers

    Match the following shapes with their properties:

    <p>Square = Area = side length² Cube = Volume = edge length³ Rectangle = Area = length × width Rectangular Prism = Volume = length × width × height</p> Signup and view all the answers

    The number 25 is not a perfect square.

    <p>False (B)</p> Signup and view all the answers

    What two factors, when multiplied, result in 25?

    <p>5 and 5</p> Signup and view all the answers

    The number _____ is a perfect square formed by multiplying two factors of 5 together.

    <p>25</p> Signup and view all the answers

    Match the following numbers with their perfect square status:

    <p>16 = Perfect square 20 = Not a perfect square 25 = Perfect square 30 = Not a perfect square</p> Signup and view all the answers

    How many Escherichia coli bacteria can fit across the diameter of a pin with a diameter of 1 mm?

    <p>1000 (C)</p> Signup and view all the answers

    A single Escherichia coli bacterium is wider than the head of a pin.

    <p>False (B)</p> Signup and view all the answers

    What is the width of one Escherichia coli bacterium in mm?

    <p>0.001</p> Signup and view all the answers

    If the diameter of the head of a pin is 1 mm, then __________ Escherichia coli bacteria can fit across it.

    <p>1000</p> Signup and view all the answers

    Match the following measurements with their descriptions:

    <p>1 mm = Diameter of a pin 0.001 mm = Width of an Escherichia coli 1000 = Number of bacteria that fit across pin Escherichia coli = Type of bacteria found in the human intestine</p> Signup and view all the answers

    Study Notes

    Topic: Exponents and Radicals

    • Exponents: Have been used to solve problems since the time of the Babylonians (over 4,000 years ago). Used in calculating interest and bacterial growth.

    • Pythagorean Triples: Sets of three positive integers such as 3, 4, and 5 representing the sides of a right triangle (e.g., Plimpton 322).

    • Bacterial Growth: Growing bacterial populations double at regular intervals, modeled using powers with integral exponents (e.g., 2⁰, 2¹, 2²).

    • Algebraic Generalizations: Used to generalize relationships through abstract thinking. Arithmetic operations extend to powers and polynomials.

    Topic: Square Roots and Cube Roots

    • Perfect Squares: A number that results from multiplying an integer by itself. (e.g., 25 = 5 x 5).

    • Square Roots: A number that, when multiplied by itself, results in the perfect square. (e.g., √25 = 5).

    • Perfect Cubes: A number that results from multiplying an integer by itself three times. (e.g., 8 = 2 x 2 x 2).

    • Cube Roots: A number that, when multiplied by itself three times, results in the perfect cube. (e.g., ∛8 = 2).

    • Relationship between side length and area of a square: The side length of a square is related to the area by the square root of the area. (e.g., if the area is 36cm², then the side length is 6cm).

    • Relationship between edge length and volume of a cube: The edge length of a cube is related to the volume by the cube root of the volume. (e.g., if the volume is 125mm³, then the edge length is 5mm).

    Topic: Exponent Laws

    • Product of Powers: Multiplying powers with the same base: am * an = am+n

    • Quotient of Powers: Dividing powers with the same base: am/an = am-n (a ≠ 0)

    • Power of a Power: Raising a power to a power: (am)n = amn

    • Power of a Product: Raising a product to a power: (ab)m = am * bm

    • Power of a Quotient: Raising a quotient to a power: (a/b)m = am/bm (b ≠ 0)

    • Zero Exponent: Any non-zero base raised to the zero power equals 1: a⁰ = 1 (a ≠ 0).

    • Negative Exponents: A negative exponent indicates a reciprocal of a power: a-n = 1/an (a ≠ 0).

    Topic: Rational Exponents

    • Rational exponent law examples: Powers with fractional or decimal exponents can be simplified using exponent rules. The denominator of the fractional exponent corresponds to the index of the radical.

    Topic: Introduction to Radicals

    • Irrational numbers: Numbers that cannot be expressed as a fraction of two integers and have non-repeating, non-terminating decimal expansions. (e.g., π, √2, √3).

    • Radical notation: Used to represent irrational numbers. (e.g., ∛8 = 2).

    • Mixed radicals: Radicals that contain a coefficient and radicand. (E.g. 2√3).

    • Entire radicals: Radicals that contain no integer coefficient. (e.g. √32).

    • Converting between powers and radicals: Fractional exponents can be converted into radical notation by making the denominator the index of the radical.

    • Converting between mixed and entire radicals: Expressing radicals as mixed or entire radicals. Finding the perfect square factors within the radical and rewriting it as product of radicals. Simlifying.

    • Ordering radicals: Ordering radicals from least to greatest involves estimating their approximate values, converting mixed radicals into entire radicals and comparing.

    Topic: Applying Exponent Laws and Radicals to Real-Life Problems

    • Solving problems using formulas: Real life examples using formulas and exponent/radical rules for exponential decay/growth, surface area calculations, volume calculations or problem-solving in relevant contexts (e.g., bacterial growth, population growth, area and volume calculations, etc).

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