Perimeter Of A Triangle With Algebraic Sides When X Equals 4

Jul 18, 2025 by ADMIN 61 views

Calculating the Perimeter of a Triangular Stage Prop

In this mathematical problem, we're presented with a scenario involving Pat, who is painting the edge of a triangular stage prop. The sides of this triangle are given in terms of algebraic expressions involving the variable 'x'. Our task is to determine the perimeter of the triangle when x equals 4. To solve this, we'll first substitute the value of x into each side's expression, then sum the resulting lengths to find the perimeter. This exercise combines algebraic substitution with geometric principles, providing a practical application of mathematical concepts.

Problem Breakdown

The lengths of the edges of the triangular stage prop are given as follows:

Side 1: $(3x - 4)$ feet
Side 2: $(x^2 - 1)$ feet
Side 3: $(2x^2 - 15)$ feet

We are asked to find the perimeter of the triangle when $x = 4$ .

Step 1: Substitute $x = 4$ into the expressions for each side

To find the length of each side, we will substitute $x = 4$ into the given expressions:

Side 1: Replace x with 4 in the expression $(3x - 4)$ . This gives us $(3(4) - 4)$ . Simplifying further, we multiply 3 by 4 to get 12, and then subtract 4. So, $12 - 4$ equals 8. Therefore, the length of the first side is 8 feet. This step involves basic arithmetic operations and demonstrates the substitution process in algebra. The ability to accurately substitute values into expressions is crucial for solving many mathematical problems, especially in algebra and calculus. It's also important to remember the order of operations (PEMDAS/BODMAS) to ensure correct calculations.
Side 2: For the second side, the expression is $(x^2 - 1)$ . Again, we substitute x with 4, resulting in $(4^2 - 1)$ . First, we need to calculate $4^2$ , which means 4 multiplied by itself, giving us 16. Then, we subtract 1 from 16. So, $16 - 1$ equals 15. This means the length of the second side is 15 feet. This calculation involves understanding exponents and their role in mathematical expressions. Exponents indicate repeated multiplication, and understanding how to work with them is a fundamental skill in algebra and various branches of mathematics.
Side 3: The expression for the third side is $(2x^2 - 15)$ . We replace x with 4, giving us $(2(4^2) - 15)$ . Following the order of operations, we first calculate $4^2$ , which is 16. Then, we multiply 16 by 2, resulting in 32. Finally, we subtract 15 from 32. So, $32 - 15$ equals 17. Therefore, the length of the third side is 17 feet. This step combines the understanding of exponents with multiplication and subtraction, further illustrating the importance of following the order of operations to arrive at the correct answer.

Step 2: Calculate the perimeter

The perimeter of a triangle is the sum of the lengths of its three sides. Now that we have the lengths of each side when $x = 4$ , we can calculate the perimeter:

Perimeter = Side 1 + Side 2 + Side 3
Perimeter = 8 feet + 15 feet + 17 feet

Adding these lengths together, we get:

Perimeter = 40 feet

Therefore, the perimeter of the triangle when $x = 4$ is 40 feet. This final step emphasizes the geometric concept of perimeter and its calculation. Understanding how to calculate the perimeter of different shapes is a basic skill in geometry and has practical applications in various real-world scenarios, such as fencing a yard or framing a picture.

Final Answer

The perimeter of the triangle is 40 feet when $x = 4$ .

Therefore, the correct answer is B. 40 feet.

Key Concepts Revisited

This problem effectively combines several key mathematical concepts. Let's revisit these to reinforce our understanding:

Algebraic Substitution

Algebraic substitution is a fundamental technique where we replace variables in an expression with specific values. In this problem, we substituted the value of x (which was 4) into the expressions representing the sides of the triangle. This allowed us to convert algebraic expressions into numerical values, which we could then use to calculate the perimeter. Mastering substitution is crucial for solving equations, evaluating functions, and simplifying complex expressions in algebra and beyond.

To further illustrate the importance of algebraic substitution, consider scenarios beyond simple geometric problems. In physics, you might substitute values for variables like velocity and time into equations of motion to calculate distance. In economics, substitution might be used to determine the cost of production based on the price of raw materials and labor. The ability to accurately substitute values is a cornerstone of quantitative reasoning in various fields.

Order of Operations (PEMDAS/BODMAS)

Following the order of operations is essential for accurate calculations. The acronyms PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) and BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction) provide a mnemonic for this order. In our problem, we encountered expressions with exponents, multiplication, and subtraction. Adhering to the order of operations ensured that we calculated the correct values for each side of the triangle. For example, we first calculated the exponent (4 squared) before performing multiplication or subtraction.

The order of operations is not merely a mathematical convention; it's a logical framework that ensures consistent and unambiguous interpretation of mathematical expressions. Without a standardized order, the same expression could yield different results, leading to confusion and errors. Consider a simple example: 2 + 3 * 4. If we perform addition before multiplication, we get 5 * 4 = 20. But if we follow the order of operations (multiplication first), we get 2 + 12 = 14. The correct answer is 14, highlighting the critical role of PEMDAS/BODMAS.

Perimeter of a Triangle

The perimeter of a triangle is the total distance around its exterior, found by adding the lengths of its three sides. In this problem, we calculated the lengths of the sides by substituting the value of x and then summed these lengths to find the perimeter. Understanding perimeter is a basic geometric concept with practical applications in measuring distances, calculating fencing requirements, and determining the amount of material needed for various projects.

Beyond triangles, the concept of perimeter extends to all polygons and curved shapes. The perimeter of a rectangle, for instance, is twice the sum of its length and width. The circumference of a circle, which is its perimeter, is calculated using the formula 2πr, where r is the radius. Understanding perimeter is crucial in fields like architecture, construction, and landscape design, where accurate measurements are essential for planning and execution.

Geometric Applications of Algebra

This problem demonstrates how algebra can be applied to geometric problems. By using algebraic expressions to represent the sides of the triangle, we were able to solve for unknown lengths and ultimately calculate the perimeter. This connection between algebra and geometry is fundamental in mathematics and allows us to solve a wide range of problems involving shapes, sizes, and spatial relationships.

The interplay between algebra and geometry is a cornerstone of mathematical problem-solving. Coordinate geometry, for example, uses algebraic equations to describe geometric shapes and analyze their properties. Calculus, a more advanced branch of mathematics, relies heavily on the fusion of algebraic and geometric concepts to study rates of change and accumulation. This problem serves as a microcosm of the broader applicability of mathematical tools in understanding and modeling the world around us.

By understanding these concepts and their applications, you can approach similar problems with confidence and accuracy. Remember to break down complex problems into smaller steps, apply the correct mathematical principles, and always double-check your work.

Practice Problems

To further solidify your understanding of these concepts, try solving the following practice problems:

The sides of a triangle are given by the expressions $(2x + 1)$ , $(x^2 - 3)$ , and $(3x^2 - 7)$ . Find the perimeter of the triangle when $x = 3$ .
A rectangle has a length of $(4y + 2)$ and a width of $(y^2 - 1)$ . Find the perimeter of the rectangle when $y = 2$ .
The perimeter of a square is given by the expression $(16z + 8)$ . Find the length of one side of the square when $z = 1$ .

By working through these practice problems, you'll gain valuable experience in applying the concepts of algebraic substitution, order of operations, perimeter calculation, and the connection between algebra and geometry.