Question

Area of the region formed by \(4|\mathrm{x}|+3|\mathrm{y}|=12\) is (1) 18 sq units (2) 20 sq units (3) 24 sq units (4) 36 sq units

Short answer

Answer: The area of the region formed by the given equation is 24 sq units.

Step-by-step solution

Identify the shape formed by the equation

The given equation is \(4|\mathrm{x}|+3|\mathrm{y}|=12\). To understand the shape formed by this equation, let's consider individual cases depending on the signs of x and y.

Divide the equation into cases

There are four cases: Case 1: x ≥ 0, y ≥ 0. In this case, the equation becomes \(4\mathrm{x} + 3\mathrm{y} = 12\). Case 2: x ≥ 0, y < 0. In this case, the equation becomes \(4\mathrm{x} - 3\mathrm{y} = 12\). Case 3: x < 0, y ≥ 0. In this case, the equation becomes \(-4\mathrm{x} + 3\mathrm{y} = 12\). Case 4: x < 0, y < 0. In this case, the equation becomes \(-4\mathrm{x} -3\mathrm{y} = 12\).

Find the coordinates of the vertices

In order to find the coordinates of the vertices, we need to find the intersection of the lines that correspond to each case. From cases 1 and 2: 1. \(4\mathrm{x} + 3\mathrm{y} = 12\) 2. \(4\mathrm{x} - 3\mathrm{y} = 12\) Subtract 2 from 1 to find y: \(6\mathrm{y} = 0 \Rightarrow \mathrm{y} = 0\) Substitute this value in case 1: \(4\mathrm{x} + 3(0) = 12 \Rightarrow \mathrm{x}=3\) The coordinates of the first vertex are \((3, 0)\). Similarly, the intersection of cases 2 and 4 gives the vertex \((-3, 0)\), the intersection of cases 3 and 4 gives the vertex \((0, -4)\), and the intersection of cases 1 and 3 gives the vertex \((0, 4)\). This gives us the vertices \((0,4)\), \((0,-4)\), \((3,0)\), and \((-3,0)\).

Calculate the area of the shape

Now that we have the vertices of the shape, we can notice that the shape formed by these vertices is a rhombus with diagonals of 6 and 8 units. Let's calculate the area of this rhombus using the formula for the area based on the length of the diagonals: Area of rhombus = \(\frac{1}{2} \times d_1 \times d_2\) Where \(d_1\) and \(d_2\) are the lengths of the diagonals. Area of rhombus = \(\frac{1}{2} \times 8 \times 6\) Area of rhombus = \(24 \thinspace sq.units\) Answer: The area of the region formed by the given equation is (3) 24 sq units.

Key concepts

Absolute Value Equations

Understanding absolute value equations is crucial when it comes to solving various mathematical problems, especially those involving distances and geometric calculations. The absolute value of a number refers to its distance from zero on the number line, without considering the direction. This distance is always a non-negative number. For instance, the absolute value of both \(3\) and \( -3\) is \(3\).

An absolute value equation includes an expression set within absolute value bars \( |x| \) and is equal to a positive number. To solve such equations, like \(4|x| + 3|y| = 12\), we consider two cases for each term: one where the term inside the bars is positive \(x \geq 0\) and one where it is negative \(x < 0\). This generates different linear equations that help determine the geometric shape formed on the coordinate plane.

Coordinate Geometry

Coordinate geometry, or analytic geometry, allows us to describe geometric shapes using algebra and numbers. This field of mathematics provides a link between algebra and geometry through graphs of lines and curves. It involves plotting points, lines, and curves on an \(xy\)-plane.

In the context of solving the textbook problem, coordinate geometry is used to find the vertices of the shape described by the absolute value equations. By considering the different cases for \(x\) and \(y\) based on their signs, we obtain linear equations whose intersections give us precise coordinates on the plane. These coordinates are crucial for further calculations, such as determining the area of geometric figures.

Area Calculation

Area calculation is a fundamental aspect of geometry that deals with finding the size of a two-dimensional shape or surface. It is commonly measured in square units. For various geometric shapes, there are different formulas to calculate their respective areas.

In this exercise, we translate the abstract equations into a physical shape using coordinate geometry. Once we have identified the shape – in this case, a rhombus – we can proceed with the appropriate formula for area calculation. The understanding of how to compute the area is not just a mathematical skill but also applies to numerous real-world situations, such as calculating the space needed for a garden or the size of a house.

Rhombus Area Formula

A rhombus is a type of polygon that is a quadrilateral with all sides having equal length. One of the properties of a rhombus is that it has diagonals that bisect each other at right angles. To find the area of a rhombus, we use the formula:\[ \text{Area of rhombus} = \frac{1}{2} \times d_1 \times d_2 \]where \(d_1\) and \(d_2\) are the lengths of the diagonals. It's important to note that the diagonals of a rhombus are perpendicular to each other, and the area is half the product of these diagonals. Understanding this formula is essential for solving geometric problems involving rhombuses, as seen in the textbook exercise where the diagonals were calculated to be 6 and 8 units, leading to the calculation of the area as 24 square units.