Finding The Side Length Of A Pentagon Sandbox

Nov 14, 2025 by Admin 46 views

Hey math enthusiasts! Let's dive into a fun geometry problem. Imagine Joan is building a sandbox – how cool is that? – and it's not just any sandbox, but one shaped like a regular pentagon. Now, we're given some details, and we need to figure out the length of one side of this awesome sandbox. Let's break this down step by step and make sure we completely understand the concept, so you guys can ace this type of question anytime.

Understanding the Problem: The Pentagon and Its Perimeter

Okay, so the core of our problem revolves around a regular pentagon. A regular pentagon is a five-sided shape where all the sides have the same length, and all the interior angles are equal. Think of it like a perfectly symmetrical house with five walls. Now, the perimeter is the total distance around the outside of the shape, just like walking all the way around the sandbox. In this case, Joan's sandbox has a perimeter of $35y^4 - 65x^3$ inches. Our goal? To find the length of a single side.

Since it's a regular pentagon, all five sides are exactly the same length. So, if we know the total perimeter, and we know there are five equal sides, we just need to divide the perimeter by 5 to find the length of one side. This is like sharing a pizza equally among five friends – you divide the whole pizza (the perimeter) by the number of friends (the sides) to see how much each person (each side) gets.

So, the main concepts here are regular pentagons, perimeter, and the idea that all sides are equal. You also need to understand how to handle algebraic expressions, particularly how to divide a binomial by a constant. This problem really is about understanding how shapes work and how their measurements relate to each other. Don't worry, once you get the hang of it, it's pretty straightforward, and you'll be able to solve similar problems without breaking a sweat. Make sure you fully understand the definitions and the main points, and you'll be on the right track!

Solving for the Side Length: A Step-by-Step Guide

Alright, time to get our hands dirty and actually solve the problem. We know the perimeter is $35y^4 - 65x^3$ inches, and we need to divide this by 5 to get the length of one side. Here’s how we'll do it:

Divide Each Term: Because we have two terms ( $35y^4$ and $-65x^3$ ), we need to divide each of them by 5 separately. This is a crucial step to remember in algebraic division. Think of it like distributing the division across both parts of the perimeter. This will make the math a lot easier, and you'll also be less likely to make a mistake.
Divide the First Term: Divide $35y^4$ by 5. The number part ( $35$ ) divided by 5 is 7. The variable part ( $y^4$ ) stays the same since there's no other y term to interact with. So, $35y^4 / 5$ becomes $7y^4$ . This part is done, and now we will go to the other side.
Divide the Second Term: Divide $-65x^3$ by 5. The number part ( $-65$ ) divided by 5 is $-13$ . The variable part ( $x^3$ ) stays the same. So, $-65x^3 / 5$ becomes $-13x^3$ . Remember, the minus sign is important here, it changes the whole thing! If you have trouble remember that a negative number divided by a positive number is a negative number, and a positive number divided by a positive number is a positive number. Also, there are no variables on the denominator so they stay on the numerator.
Combine the Results: Now, put the results from steps 2 and 3 together. We have $7y^4$ and $-13x^3$ . Combine them, and you get $7y^4 - 13x^3$ . This is the expression for the length of one side of the pentagon.

And that's it! You have successfully calculated the length of one side of the sandbox, by dividing the perimeter by 5. Make sure you take your time, and write down your steps, this will help you in every math problem.

Analyzing the Answer Choices: Finding the Right Match

Now that we've found the side length ( $7y^4 - 13x^3$ ), let's look at the answer choices provided in the problem. We want to find the one that matches our calculated result. In the options, it seems like there might be a small mistake in the original question: We correctly found that one side is $7y^4 - 13x^3$ inches. Now, let's look at the options:

A. $5y - 9$ inches: This doesn't match our answer at all. The variables and coefficients are completely different.
B. $5y^4 - 9x^3$ inches: This also does not match our calculated answer, the values are different.
C. $7y^4 - 13x^3$ inches: Bingo! This matches our calculated answer perfectly. The coefficients (7 and -13) and the variables with their exponents ( $y^4$ and $x^3$ ) are exactly what we found.

So, the correct answer is C. You did it! You've successfully navigated through the problem, calculated the side length, and correctly identified the matching answer choice. This is the moment to congratulate yourself! You've shown that you understand the concepts of perimeter, regular polygons, and basic algebraic division, all within the context of a fun, real-world example.

Key Takeaways: Mastering the Problem

To recap, here are the key takeaways from this problem:

Understanding the Pentagon: A regular pentagon has five equal sides, and all the interior angles are the same.
Perimeter Basics: The perimeter is the total distance around a shape. For a regular pentagon, it's the sum of the lengths of all five sides.
Division in Algebra: When you divide a polynomial (like our perimeter) by a number, you divide each term of the polynomial separately.
Matching the Answer: Always make sure you compare your calculated answer to the available options, carefully checking both the coefficients and the variables.

This kind of problem helps you see how math is connected to everyday things. You used your math skills to figure out the size of a sandbox – pretty cool, right? Keep practicing similar problems to get better at recognizing the patterns and applying the right formulas. The more you practice, the easier it will become, and soon you'll be solving these problems without a hitch. Remember to always double-check your work and to focus on understanding the underlying concepts, not just memorizing formulas. You got this, guys!

Expanding Your Knowledge: Similar Problems and Concepts

Let's keep the momentum going. Now that you've tackled this pentagon sandbox problem, you can apply the same concepts to other geometry problems. Here are some related concepts and problem types to try out:

Other Regular Polygons: What if Joan built a sandbox in the shape of a hexagon (6 sides), an octagon (8 sides), or even a decagon (10 sides)? The principle remains the same: If it's a regular polygon, all sides are equal. You'd divide the perimeter by the number of sides.
Finding the Perimeter from a Side Length: Instead of being given the perimeter, you might be given the side length and asked to find the perimeter. For a regular polygon, you'd simply multiply the side length by the number of sides.
Area Problems: Once you're comfortable with perimeter, you could also explore finding the area of the pentagon or other polygons. This involves different formulas, but the basic geometric principles still apply.
Algebraic Expressions: Practice working with more complex algebraic expressions. This will make it easier to understand problems where the measurements are given in terms of variables like x and y.

By practicing these kinds of problems, you'll strengthen your skills in both geometry and algebra. Keep exploring, keep practicing, and don't be afraid to try new things. Math is all about building upon what you already know, so take these new challenges and see what you can do. You will do a great job. Also, remember, it's about the process, not just the answer. Always show your work, and don't hesitate to ask for help if you need it. You can do this!