Joan is building a sandbox in the shape of a regular pentagon. the perimeter of the pentagon is 35y4 – 65×3 inches. what is the length of one side of the sandbox?
The Correct Answer and Explanation is:
To find the length of one side of a regular pentagon with a perimeter of (35y^4 – 65x^3) inches, we can divide the perimeter by 5, since a regular pentagon has five equal sides.
Step-by-Step Solution
- Given perimeter: The perimeter of the pentagon is (35y^4 – 65x^3).
- Number of sides: A pentagon has five sides, so each side will be one-fifth of the total perimeter.
- Dividing the expression: We divide the perimeter expression by 5 to find the length of one side: [
\text{Length of one side} = \frac{35y^4 – 65x^3}{5}
] - Simplify each term:
- For the term (35y^4): (\frac{35y^4}{5} = 7y^4).
- For the term (-65x^3): (\frac{-65x^3}{5} = -13x^3). Therefore: [
\text{Length of one side} = 7y^4 – 13x^3
]
Final Answer
The length of one side of the sandbox is (7y^4 – 13x^3) inches.
Explanation
In geometry, a regular polygon is a shape with all sides and angles equal. For a regular pentagon, this means each of its five sides has the same length. When given the perimeter of a regular polygon, we can determine the length of one side by dividing the total perimeter by the number of sides. This works because dividing evenly ensures each segment represents a side length.
In this case, the algebraic expression (35y^4 – 65x^3) represents the perimeter in terms of variables. Dividing each term of this expression by 5 distributes the calculation across each term, resulting in (7y^4 – 13x^3), which is the length of one side.
This process of dividing and simplifying terms applies to any algebraic perimeter of a regular polygon; each side length will be the perimeter divided by the number of sides, simplified term-by-term.