Geometry: Volume and Properties of a Cube

A flat shape with four equal sides and four right angles.

The space inside a two-dimensional figure.

The length of one side of a square.

A three-dimensional shape with six equal square faces.

The length of an edge of a cube.

The relationship between the side length and the area of a square is that the area is always the side length squared.

For any square, if you know the side length, you can easily calculate the area by squaring the side length. And if you know the area, you can find the side length by taking the square root of the area.

The amount of space a three-dimensional object occupies.

The length of a side of a cube.

The cube root of a number is the value that, when multiplied by itself three times, equals the original number.

Finding the value of the edge of a cube when you know its volume.

The distance between two opposite vertices of a cube, passing through the center of the cube.

The amount of space a cube occupies, calculated by cubing the side length (V = s³).

The length of one edge of a cube, used to calculate its volume and surface area.

To multiply a number by itself three times (n x n x n).

A three-dimensional shape with six equal square faces, 12 edges, and 8 vertices.

A number or symbol that indicates how many times a base number or expression is multiplied by itself.

The number or expression that is being multiplied by itself.

A negative exponent means you take the reciprocal of the base raised to the positive version of the exponent; Example: 4^-3 = 1/(4^3)

The process of repeatedly multiplying a number by itself a certain number of times, determined by the exponent.

A mathematical symbol representing infinity, often used to express unbounded growth or limits.

Expressing a number as a product of a base multiplied by itself a certain number of times.

The number that is being multiplied by itself.

Writing a number as a power of 10, which means the base number is 10 and the exponent indicates the number of zeros in the original number.

A number that can be expressed as a power of 10 with a positive exponent.

The length of one side of a square.

The area of a square is calculated by multiplying the side length by itself.

The length of an edge of a cube.

The volume of a cube is calculated by multiplying the edge length by itself three times.

A perfect square is a number that can be obtained by squaring an integer. In other words, it's the result of multiplying an integer by itself.

25 is a perfect square because it's the result of multiplying 5 by itself (5 * 5 = 25).

Finding the square root of a number means finding the value that, when multiplied by itself, equals the original number. For example, the square root of 25 is 5, because 5 * 5 = 25.

The square root of a perfect square is an integer. For example, the square root of 25 is 5, which is an integer.

Perfect squares are found by multiplying an integer by itself. This means you can get a perfect square by squaring any whole number.

A NiMH battery, commonly found in power tools, loses about 30% of its stored energy every month if left uncharged.

The width of a single Escherichia coli bacterium is 10⁻³ mm, while the diameter of a pinhead is 1 mm. To find out how many bacteria fit across the pinhead, we need to calculate the ratio of the pinhead's diameter to the bacterium's width.

To find the number of bacteria that can fit across the pinhead, we divide the pinhead's diameter (1 mm) by the bacterium's width (10⁻³ mm).

Nickel-metal hydride (NiMH) is a type of rechargeable battery used in many devices.

1 mm / 10⁻³ mm = 1000. This means 1000 Escherichia coli bacteria can fit across the diameter of a pinhead.

A NiMH battery's charge will decline naturally over time, even when not in use.

When working with very large or very small numbers, scientific notation provides a convenient way to express them.

The rate at which a NiMH battery's charge depletes without use is roughly 30% per month.

To prevent significant charge loss, regularly recharge NiMH batteries used in power tools.

1000 can be written in scientific notation as 1 x 10³ or simply 10³.

A perfect cube is a number that can be obtained by cubing an integer (multiplying an integer by itself three times). Example: 27 is a perfect cube because 3 x 3 x 3 = 27.

A number that is both a perfect square and a perfect cube is a number that can be obtained by squaring an integer and also by cubing another integer. For example, 64 is both a perfect square (8 x 8 = 64) and a perfect cube (4 x 4 x 4 = 64).

To find the square root of a number, you need to find a number that, when multiplied by itself, equals the original number. For example, the square root of 25 is 5 because 5 x 5 = 25. To find the cube root of a number, you need to find a number that, when multiplied by itself three times, equals the original number. For example, the cube root of 8 is 2 because 2 x 2 x 2 = 8.

Factor trees are a way to break down a number into its prime factors. Each number is factored into two smaller numbers until you're left with only prime numbers. This helps you find the prime factorization of the number. This can help find prime factors for both squares and cubes.

The formula C = (1/21)t describes the fraction of a pond's surface covered by algae, where C represents the fraction and t is the number of weeks ago.

To find when 25% of the pond was covered, we need to solve the equation C = (1/21)t for t, where C = 0.25 (representing 25%).

In the context of the pond, time is measured in weeks. The formula C = (1/21)t tells us the fraction of the pond's surface area covered by algae t weeks ago.

The problem asks for the time when 25% of the pond was covered by algae. This means we need to find the value of t when C = 0.25. We'll substitute 0.25 for C in the formula and solve for t.