Finding The Side Length Of A Pentagon Sandbox

by Alex Johnson 46 views

Hey there, math enthusiasts! Today, we're diving into a fun geometry problem involving Joan and her sandbox. Joan is constructing a sandbox, and it's not just any shape – it's a regular pentagon! Let's break down the problem step-by-step to find the length of one side of the sandbox. This is a common type of problem that combines geometric shapes with algebraic expressions, which is a great way to practice your skills.

Understanding the Problem: The Pentagon and Its Perimeter

First things first, what's a regular pentagon? Well, it's a five-sided polygon where all sides have the same length, and all interior angles are equal. This is a crucial piece of information because it tells us that all the sides of Joan's sandbox are identical. Now, the problem gives us the perimeter of the pentagon. The perimeter is the total distance around the outside of the shape, or the sum of the lengths of all its sides. In this case, the perimeter is expressed as the algebraic expression 35y4−65x335y^4 - 65x^3 inches. Our goal is to figure out the length of one side, knowing the total perimeter. To do this, we'll need to use some basic algebraic principles.

Now, let's connect the dots. We know the perimeter and that the pentagon has five equal sides. That means we need to divide the total perimeter by the number of sides to find the length of one side. Think of it like this: if you have five identical objects and their total weight is known, you find the weight of one object by dividing the total weight by five. The same principle applies here. The perimeter is a sum, and we'll undo that with division. Let’s get to work!

Solving for the Side Length: Algebraic Division

Here’s how we'll do it. Since a pentagon has five sides, we'll divide the perimeter, 35y4−65x335y^4 - 65x^3, by 5. This will give us the length of one side. The key here is to divide each term in the expression by 5. Here's how the math looks:

rac{35y^4 - 65x^3}{5}

Now, let's break it down term by term:

  • Divide 35y435y^4 by 5: rac{35y^4}{5} = 7y^4
  • Divide −65x3-65x^3 by 5: rac{-65x^3}{5} = -13x^3

So, putting it all together, the expression becomes: 7y4−13x37y^4 - 13x^3. This expression represents the length of one side of the pentagon sandbox in inches. This is a classic example of applying algebraic principles to geometric shapes, showing how the two areas of mathematics are frequently intertwined. Now, let’s go back to our multiple-choice options to identify the correct answer.

Analyzing the Answer Choices

Now that we've solved the problem, let's look at the answer choices provided. We're looking for an answer that matches our calculated side length, which is 7y4−13x37y^4 - 13x^3 inches.

  • A. 5y−95y - 9 inches: This option is incorrect because it uses different coefficients and variables than our calculated answer.
  • B. 5y4−9x35y^4 - 9x^3 inches: This option is also incorrect, as the coefficients are different from our solution.
  • C. 7y−137y - 13 inches: This option is incorrect as it has the correct coefficients but the variables have the wrong exponents.
  • D. 7y4−13x37y^4 - 13x^3 inches: This is the correct answer! It perfectly matches the side length we calculated by dividing the perimeter by 5. The coefficients and the exponents of the variables are identical to our solution.

So, the correct answer is D. This shows a good understanding of algebraic expressions, geometric shapes, and how to apply division to find missing lengths. Always remember to break down the problem into smaller, manageable steps.

Conclusion: Joan's Sandbox Solution!

We successfully found the length of one side of Joan's pentagon sandbox. We started with the perimeter, understood the properties of a regular pentagon, and used algebraic division to find our answer. The key takeaway here is the ability to connect geometric concepts with algebraic expressions. Problems like this are great practice for both areas and help build a stronger mathematical foundation. Always make sure to double-check your work, and don't hesitate to break down problems into smaller steps! Keep practicing and exploring, and you'll find that math can be both fun and rewarding.

In essence, you took a geometric problem involving the perimeter of a pentagon and used algebraic division to find the length of one side. This is a common application of math principles, reinforcing the importance of understanding the relationship between different branches of mathematics. By breaking down the problem into smaller steps, you made the solution more accessible and easier to understand. Excellent job!

If you want to delve deeper, you can also explore how to calculate the area of a pentagon given the side length or look at how the perimeter changes if you change the side lengths. Keep up the great work, and happy solving!

For further learning, check out resources on geometry and algebra on websites like Khan Academy. They offer a wealth of lessons and practice problems to hone your skills in both areas.