Perimeter Of A Triangle Stage Prop A Mathematical Exploration

Introduction

In this mathematical exploration, we delve into the fascinating world of geometry and algebra to determine the perimeter of a triangular stage prop. Kat, the protagonist of our problem, is diligently painting the edges of this triangular masterpiece with vibrant reflective orange paint. The lengths of the triangle's edges are expressed as algebraic expressions: $(3x - 4)$ feet, $(x^2 - 1)$ feet, and $(2x^2 - 15)$ feet. Our mission, should we choose to accept it, is to calculate the perimeter of this triangle when $x$ is equal to 4. This involves substituting the value of $x$ into the expressions, calculating the side lengths, and finally, summing them up to find the perimeter. This exercise not only reinforces our understanding of algebraic substitution and geometric principles but also showcases how mathematics plays a practical role in real-world scenarios, such as stage design and prop construction. So, let's embark on this mathematical journey and uncover the perimeter of Kat's triangular stage prop.

Understanding the Problem

To effectively tackle this problem, we must first grasp the fundamental concepts involved. At its core, this problem requires us to calculate the perimeter of a triangle. The perimeter, in geometric terms, refers to the total distance around the outside of a two-dimensional shape. In the case of a triangle, it's simply the sum of the lengths of its three sides. Now, the lengths of these sides are not given as fixed numerical values but rather as algebraic expressions. This adds an interesting twist to the problem, as we need to employ our algebraic skills to find the actual side lengths. We are given that the side lengths are $(3x - 4)$ feet, $(x^2 - 1)$ feet, and $(2x^2 - 15)$ feet. The variable $x$ plays a crucial role here, and we are provided with its value: $x = 4$. This is where the algebraic substitution comes into play. We will substitute this value of $x$ into each of the expressions representing the side lengths. Once we've done that, we'll have the numerical lengths of the three sides, and calculating the perimeter will be a straightforward addition task. The beauty of this problem lies in its integration of algebra and geometry, showcasing how these mathematical branches intertwine to solve practical problems.

Step-by-Step Solution

Now, let's meticulously walk through the solution process, step by step, to ensure clarity and understanding. Our primary goal is to determine the perimeter of the triangle when $x = 4$. To achieve this, we'll break down the problem into manageable steps:

1. Substitute the value of $x$ into each expression representing the side lengths:

   • Side 1: $3x - 4$
   • Side 2: $x^2 - 1$
   • Side 3: $2x^2 - 15$

2. Calculate the numerical value of each side length by performing the arithmetic operations after substitution.

3. Add the three side lengths together to find the perimeter of the triangle.

Let's begin with the first side, which is expressed as $3x - 4$. Substituting $x = 4$ into this expression, we get:

$3(4) - 4 = 12 - 4 = 8$ feet

So, the length of the first side is 8 feet.

Next, we move on to the second side, represented by the expression $x^2 - 1$. Substituting $x = 4$ here gives us:

$(4)^2 - 1 = 16 - 1 = 15$ feet

Therefore, the second side of the triangle measures 15 feet.

Finally, we consider the third side, which is expressed as $2x^2 - 15$. Substituting $x = 4$ into this expression, we get:

$2(4)^2 - 15 = 2(16) - 15 = 32 - 15 = 17$ feet

Thus, the length of the third side is 17 feet.

Now that we have the lengths of all three sides, we can calculate the perimeter by adding them together:

Perimeter = Side 1 + Side 2 + Side 3

Perimeter = 8 feet + 15 feet + 17 feet

Perimeter = 40 feet

Therefore, the perimeter of the triangular stage prop is 40 feet.

Detailed Calculations

To ensure absolute clarity and leave no room for doubt, let's delve into a more detailed breakdown of the calculations involved in determining the side lengths of the triangle. This will not only solidify our understanding but also provide a valuable reference for future similar problems. We'll revisit each side length expression and meticulously perform the substitution and arithmetic operations.

Side 1: $3x - 4$

As we established earlier, the first side is represented by the expression $3x - 4$. Our task is to substitute $x = 4$ into this expression and simplify it to obtain the numerical length of the side. Let's break it down:

1. Substitution: Replace $x$ with 4 in the expression: $3(4) - 4$

2. Multiplication: Perform the multiplication operation: $12 - 4$

3. Subtraction: Perform the subtraction operation: $8$

Therefore, the length of the first side is 8 feet.

Side 2: $x^2 - 1$

The second side is represented by the expression $x^2 - 1$. Again, we substitute $x = 4$ and simplify:

1. Substitution: Replace $x$ with 4 in the expression: $(4)^2 - 1$

2. Exponentiation: Evaluate the exponent: $16 - 1$

3. Subtraction: Perform the subtraction operation: $15$

Hence, the length of the second side is 15 feet.

Side 3: $2x^2 - 15$

The third side is represented by the expression $2x^2 - 15$. Substituting $x = 4$ and simplifying:

1. Substitution: Replace $x$ with 4 in the expression: $2(4)^2 - 15$

2. Exponentiation: Evaluate the exponent: $2(16) - 15$

3. Multiplication: Perform the multiplication operation: $32 - 15$

4. Subtraction: Perform the subtraction operation: $17$

Thus, the length of the third side is 17 feet.

By meticulously breaking down each calculation, we have reinforced our understanding of the substitution process and the order of operations. This detailed approach ensures accuracy and clarity in our solution.

Final Answer

After meticulously substituting the value of $x = 4$ into the expressions representing the side lengths of the triangular stage prop, we have determined the numerical lengths of each side. The first side measures 8 feet, the second side measures 15 feet, and the third side measures 17 feet. To find the perimeter, we simply add these side lengths together:

Perimeter = 8 feet + 15 feet + 17 feet = 40 feet

Therefore, the perimeter of the triangular stage prop is 40 feet. This means that Kat will need enough reflective orange paint to cover a total of 40 feet along the edges of the triangle. Our journey through algebra and geometry has culminated in a clear and concise answer, demonstrating the power of mathematical principles in solving real-world problems. We have not only found the perimeter but also reinforced our understanding of algebraic substitution and geometric concepts.

Conclusion

In conclusion, we have successfully navigated the mathematical landscape of this problem, starting with algebraic expressions and culminating in a numerical answer. We embarked on a journey to determine the perimeter of a triangular stage prop, guided by the principles of algebra and geometry. Through careful substitution, meticulous calculations, and a clear understanding of the problem, we arrived at the solution: the perimeter of the triangle is 40 feet. This exercise has not only provided us with a concrete answer but has also underscored the interconnectedness of mathematical concepts. We saw how algebra, with its variables and expressions, seamlessly blends with geometry, which deals with shapes and their properties. The ability to translate real-world scenarios into mathematical models and solve them is a valuable skill, and this problem serves as a testament to the practical applications of mathematics. As Kat diligently paints the edges of her triangular masterpiece, we can appreciate the mathematical precision that underlies her artistic endeavor. The perimeter, a simple yet fundamental geometric concept, becomes a tangible measure of the length Kat needs to paint, bridging the gap between abstract mathematics and the concrete world.