devyy A rational number is defined as any number that can be expressed as the ratio of two integers, where the numerator is an integer and the denominator is a non-zero integer. In simpler terms, it can be written in the form p/q, with p and q being integers and q ≠ 0. Rational numbers can … Read more
When discussing the properties of trapezoids, many people might think that the median is simply the average of the lengths of the two bases. While it’s easy to see why one might arrive at this conclusion, it’s crucial to clarify what the median of a trapezoid actually represents. The median of a trapezoid is defined … Read more
Geometrical Illustrations In this section, we will illustrate four different geometrical scenarios, each accompanied by a rough figure and appropriate labels. 1. A Point P on Line Segment AB A B P In the above figure, point P lies directly on line segment AB. 2. Intersection of XY and PQ at M X Y P … Read more
To find the volume of the solid that lies beneath the plane 3x + 2y + z = 12 and above the rectangle R defined by 0 ≤ x ≤ 1 and 3 ≤ y ≤ 4, follow these steps: Step 1: Rearrange the Plane Equation First, we rearrange the equation of the plane to … Read more
To solve the system of equations given by 2xy = 0 and 3xy = 9, we need to analyze each equation individually. Step 1: Analyze the first equation The first equation, 2xy = 0, implies that either x = 0, y = 0, or both. This means all points along the x-axis (y = 0) … Read more
Solving the System of Equations To solve the system of equations given by: Equation 1: xy = 12 Equation 2: x² + y² = 40 Step 1: Express one variable in terms of the other From the first equation, we can express y in terms of x: y = 12/x Step 2: Substitute into the … Read more
To find the value of the expression 2x² + 3xy + 4y² when x = 2 and y = 4, we will substitute the values of x and y into the expression. First, substitute x and y: 1. Calculate 2x²: 2(2)² = 2(4) = 8 2. Calculate 3xy: 3(2)(4) = 3(8) = 24 3. Calculate … Read more
In a perfectly symmetrical distribution, such as a normal distribution, the mean, median, and mode are all equal. Therefore, if you have a symmetrical distribution with a mean (μ) of 30, the mode is also 30. To elaborate, the mode is the value that appears most frequently in a data set. In cases of perfect … Read more
To find the points on the surface defined by the equation y² + 49xz = 0 that are closest to the origin, we will use the method of Lagrange multipliers. This technique finds the extrema of a function subject to a constraint. Firstly, we need to express our goal mathematically. The distance from the origin … Read more
Solving the System of Equations To solve the system of equations: Equation 1: x² + y = 3 Equation 2: x + y = 3 we can use substitution or elimination methods. Here, we will use substitution for simplicity. Step 1: Solve Equation 2 for y From Equation 2, we can express y in terms … Read more