Understanding the Relationship Between Surface Area and Volume of a Cube A cube is a three-dimensional shape where all sides are equal in length. To express the surface area of a cube as a function of its volume, we first need to recall the formulas for both the surface area and volume of a cube. … Read more
Finding the Minimum Value of the Product xy To determine the minimum value of the product xy given the equation y = 2x + 8, we can start by rewriting the product in terms of a single variable. First, replace y in the product: xy = x(2x + 8) = 2x^2 + 8x Now, we … Read more
To determine the number of sides in a regular polygon when given the measure of its exterior angle, we can use the formula: Exterior Angle = 360° / n, where n represents the number of sides of the polygon. In this case, we know that the exterior angle is 24 degrees. We can set up … Read more
A rectangle is a type of quadrilateral that possesses certain properties which qualify it as a convex shape. To understand why a rectangle is a convex quadrilateral, we first need to clarify what these terms mean. A quadrilateral is a polygon with four sides. Convex quadrilaterals, specifically, are those where all interior angles are less … Read more
To find the solution to the system of equations given by y = 4x + 4 and y = 3x + 3, we can use the substitution or elimination method. In this case, we will use the substitution method: 1. Since both equations are equal to y, we can set them equal to each other: … Read more
To find the factors of the quadratic expression m² + 14m + 48, we need to factor it into the form (m + p)(m + q), where p and q are such that: p + q = 14 (the coefficient of m) p * q = 48 (the constant term) Next, we need to identify … Read more
Adding and subtracting integers can be accomplished easily without a number line by using mental math strategies, understanding the rules of signs, and employing visual aids or manipulatives. Here’s a breakdown of some effective methods: Addition of Integers 1. **Same Signs**: When adding two integers with the same sign, simply add their absolute values and … Read more
A three-dimensional shape, often referred to as a 3D shape, is a form that has depth, width, and height, distinguishing it from two-dimensional shapes that have only length and width. These shapes occupy space and can be found all around us, from everyday objects to complex architectural designs. In geometry, three-dimensional shapes can be classified … Read more
To factor the expression 3x² x 7 completely, we begin by identifying the components of the expression. This expression consists of two factors: 3x² and 7. 1. **Identify the individual factors**: The first part of our expression, 3x², can be thought of as having its own factors: 3: This is a prime number, and it … Read more
To determine whether the numbers 24, 32, and 40 can form a right triangle, we can apply the Pythagorean theorem. The Pythagorean theorem states that for a right triangle with sides of lengths a, b, and hypotenuse c, the following equation holds: a² + b² = c² In our case, we need to identify the … Read more