Triangle Perimeter Calculation: A Step-by-Step Guide

Hey everyone! Today, we're diving into a fun geometry problem that's all about triangles and their perimeters. The main question we are trying to solve is: What is the perimeter of the triangle? We're given the lengths of a triangle's sides in terms of x, and our mission is to calculate the perimeter when x equals 4. Don't worry, it's easier than it sounds! Let's break it down step by step to make sure you get it. This guide is designed to be super clear, so anyone can follow along, whether you're a math whiz or just getting started. So, grab your pencils, and let's get started with this awesome math problem.

Understanding the Problem: Triangle Sides and Perimeter

First things first, let's make sure we're all on the same page. The problem gives us the lengths of a triangle's edges in terms of x. The sides are $(3x-4)$ feet, $(x^2-1)$ feet, and $(2x^2-15)$ feet. Remember, the perimeter of any shape is the total distance around its outside. For a triangle, that means adding up the lengths of all three sides. In our case, to find the perimeter, we need to add $(3x-4) + (x^2-1) + (2x^2-15)$. But we're not done yet, we have to find the value of the perimeter when x = 4. This means we'll substitute 4 for x in each of the expressions and then add them up. Knowing the basics of algebra and geometry is super important. We’ll be using these concepts to solve for the perimeter. Understanding this is key to solving the problem. The question wants us to find the total distance around the triangle when x equals a specific value. So, we'll start by substituting the value of x in the given expressions.

Now, let's translate the problem into a simple, step-by-step process. First, we identify the expressions representing the sides of the triangle. Second, we substitute the value of x with 4 in each of these expressions. Third, we calculate the length of each side using the substituted value. Finally, we add up all the side lengths to determine the perimeter of the triangle. Each step plays a crucial role in obtaining the correct answer. The process is straightforward, and we'll go through it bit by bit, ensuring you grasp every element of the solution. By breaking down the problem this way, it becomes much easier to manage and comprehend. We have to be meticulous in our calculations to avoid any errors. Remember, every little detail matters when dealing with mathematical equations. Let's make sure our math skills are on point!

Step-by-Step Solution: Finding the Perimeter

Alright, let's get down to the nitty-gritty and calculate the perimeter! We know that the lengths of the sides are $(3x-4)$, $(x^2-1)$, and $(2x^2-15)$, and we need to find the perimeter when x = 4. Here's how we'll do it step by step:

1. Substitute x = 4 into each expression:

   • For the first side: $(3x - 4)$ becomes $(3*4 - 4)$
   • For the second side: $(x^2 - 1)$ becomes $(4^2 - 1)$
   • For the third side: $(2x^2 - 15)$ becomes $(2*4^2 - 15)$

2. Calculate the length of each side:

   • First side: $(3*4 - 4) = (12 - 4) = 8$ feet
   • Second side: $(4^2 - 1) = (16 - 1) = 15$ feet
   • Third side: $(2*4^2 - 15) = (2*16 - 15) = (32 - 15) = 17$ feet

3. Find the perimeter by adding up the side lengths:

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

So, the perimeter of the triangle when x = 4 is 40 feet. This method is incredibly useful for solving any geometry problem that involves calculating perimeters. You start with the given information, make the necessary substitutions, simplify the expressions, and finally, add everything up to get your answer. Isn't math awesome, guys?

Detailed Calculations and Explanations

Let's go into more detail to make sure everything is crystal clear. First, we substituted the value of x into the expression representing each side of the triangle. This is a fundamental step in algebra. By doing this, we turned expressions containing variables into simple numerical values, which are much easier to work with. Specifically, when we replaced x with 4, it changed the nature of the expression, making it a concrete number. For example, $(3x - 4)$ became $(3*4 - 4)$. It's a direct substitution, with the variable being replaced by its given value.

Second, we then focused on simplifying the equations. To make sure we're doing things right, we followed the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). For instance, when we evaluated the first side, we multiplied 3 by 4 before subtracting 4. With the second side, we squared 4 before subtracting 1. And with the third side, we squared 4, multiplied by 2, and then subtracted 15. The accurate application of the order of operations ensures that our calculations are correct and that we arrive at the correct side lengths. This attention to detail is crucial for the overall solution.

Third, we found the perimeter by adding the results. This is the final step in the process, combining all the individual side lengths. Since the perimeter of the triangle is the sum of its sides, we just had to add 8 feet, 15 feet, and 17 feet. This simple addition provided us with the final answer: 40 feet. The entire process illustrates how a seemingly complex problem is made easy by breaking it down into smaller, manageable steps.

Conclusion: The Answer and What We've Learned

So, we've successfully found the perimeter of the triangle! The correct answer is B. 40 feet. We started with the side lengths of the triangle in terms of x, substituted x with 4, calculated the length of each side, and finally, added them up to find the perimeter. Isn't that cool?

What did we learn today? We brushed up on the concept of a triangle's perimeter, how to substitute variables in algebraic expressions, and how to follow the order of operations to solve equations. We also practiced breaking down a word problem into smaller steps, which makes it easier to solve. The skills you learned today are applicable in various areas of math and real life. Always remember to stay focused, take things one step at a time, and you'll do great. Keep practicing and exploring these concepts, and you'll find that math can be fun and rewarding!

This method can be used for any triangle problem. Make sure to understand the fundamental concepts and the methods that we learned today. Always follow the correct order of operation to ensure the accuracy of the answer. The more you practice, the better you'll get. That's the key to mastering any math concept.

Additional Tips and Practice Problems

To solidify your understanding and skills, let's explore some extra tips and practice problems. First, let's consider the concept of units. While this problem deals with feet, it's vital to ensure that all measurements are in the same unit. If you're given different units, convert them before you start your calculations. This prevents errors and ensures your final answer is accurate. You can also try to use different shapes with perimeters in mind, it will help you understand the concept.

Next, here's a practice problem to try on your own: The sides of a triangle are given by the expressions $(2x + 1)$ cm, $(3x - 2)$ cm, and $(x + 4)$ cm. What is the perimeter of the triangle if x = 3? Remember to follow the steps we discussed: Substitute, calculate, and add. This will test your knowledge, so give it a shot, and see if you can solve it correctly. After solving this problem, compare it with the solution. If there's something you do not understand, feel free to review the section where we discussed the step-by-step methods. If you have more questions, you can always ask your friends or your teachers, so you could solve and learn together. Make sure you understand the concepts clearly.

Remember, practice makes perfect. The more you work through problems like these, the better you'll become at solving them. Math is like any other skill. By consistently practicing, you'll improve your abilities and gain confidence in your problem-solving skills. So keep it up, keep learning, and don't be afraid to challenge yourself with more complex problems!