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To find the point on the line defined by the equation y = 4x + 5 that is closest to the origin (0, 0), we can use the concept of the distance between a point and a line. The distance d from a point (x_0, y_0) to a line in the form of Ax + … Read more

To determine the length of the side opposite the 60-degree angle in a triangle, we first need additional information about the triangle’s other sides or angles. Assuming we have a triangle, we can use the properties of special triangles or the Law of Sines. In a 30-60-90 triangle, the sides are in a specific ratio. … Read more

To determine the number of sides in a polygon when given specific angle measures, we can use the formula for the sum of interior angles of a polygon. The formula is: Sum of interior angles = (n – 2) × 180° where n represents the number of sides in the polygon. In this scenario, we … Read more

The recursive definition of the sequence a can be derived by analyzing how each term relates to the previous terms in the sequence. Here’s a detailed explanation: 1. **Base Case**: To begin the recursion, we need to provide a starting point. Let’s define the first term: a1 = 1 This means that the first term … Read more

Converting a mixed number to a fraction involves a few simple steps. A mixed number consists of a whole number and a proper fraction, such as 2 3/4. Here’s how to convert it: Multiply the whole number by the denominator: Take the whole number part (2) and multiply it by the denominator of the fraction … Read more

Finding the Area of a Sector To find the area of a sector of a circle, we can use the following formula: Area of Sector = (θ / 360) × π × r² Where: θ = angle of the sector in degrees r = radius of the circle In this case, we have: θ = … Read more

Understanding the Venn Diagram To find the values of n, a, b, and c from a Venn diagram, we first need to identify what each variable represents. Typically in Venn diagrams, n can denote the total number of elements in the sample set, while a, b, and c generally refer to the cardinalities of specific … Read more

To calculate the future value of an investment with a fixed annual interest rate, you can use the formula for compound interest: Future Value (FV) = P (1 + r)^t Where: FV: Future Value of the investment P: Principal amount (initial investment) r: Annual interest rate (as a decimal) t: Number of years the money … Read more

The phrase ‘3 more than the product of 8 and a number’ can be broken down into two parts for better clarity. 1. **Understanding the Product**: The term ‘the product of 8 and a number’ refers to multiplication. If we denote ‘a number’ as ‘x’, then the product is represented mathematically as 8 * x. … Read more

Solution to the System of Linear Equations To find the values of variables q and r from the following system of linear equations: Equation 1: 12q + 3r = 15 Equation 2: 4q + 4r = 44 let’s solve them step by step. Step 1: Simplify Equations First, we can simplify Equation 2 for better … Read more