Laura built a rectangular patio using 12 paving stones of equal length. The paving stones were each 6 feet long. Which expression has a value equal to the perimeter of the patio?
Options:
2^3 ×3^2
3^2 ×2^3
6^2 ×2^2
6^3 ×2^1

To solve this problem, let's break it down step by step:

1. **Understanding the problem**:
   - Laura used 12 paving stones, each 6 feet long, to build a rectangular patio.
   - The perimeter of a rectangle is calculated as \( P = 2 \times (\text{Length} + \text{Width}) \).

2. **Assumptions**:
   - Let's assume the patio's length is \( L \) and its width is \( W \).
   - The total length of all the paving stones combined is \( 12 \times 6 = 72 \) feet.

3. **Setting up the equations**:
   - The length and width of the patio should add up to half the total length of the paving stones used, since \( L + W \) must be half of 72 feet (as the perimeter formula accounts for two lengths and two widths).

     So, \( L + W = \frac{72}{2} = 36 \) feet.

4. **Calculating the perimeter**:
   - The perimeter of the patio would then be \( P = 2 \times 36 = 72 \) feet.

5. **Matching this with the given expressions**:
   - Let's evaluate each expression to find which one equals 72.

   \[
   \text{Option 1: } 2^3 \times 3^2 = 8 \times 9 = 72
   \]
   \[
   \text{Option 2: } 3^2 \times 2^3 = 9 \times 8 = 72
   \]
   \[
   \text{Option 3: } 6^2 \times 2^2 = 36 \times 4 = 144
   \]
   \[
   \text{Option 4: } 6^3 \times 2^1 = 216 \times 2 = 432
   \]

   Both Option 1 and Option 2 have a value of 72, which matches the perimeter.

**Answer:**
The correct expressions are **\(2^3 \times 3^2\)** and **\(3^2 \times 2^3\)**, both of which equal the perimeter of the patio.

Would you like more details on any part of this solution?

---

Here are 5 related questions:

1. How do you calculate the area of the patio if the length and width are known?
2. What would happen to the perimeter if each paving stone was only 4 feet long?
3. How would the perimeter change if Laura used 14 paving stones instead of 12?
4. Can you express the perimeter using a different mathematical expression?
5. How does changing the shape of the patio (but keeping the same total length of stones) affect the perimeter?

**Tip**: When calculating perimeters and areas, always double-check whether the problem involves a simple geometric shape or a combination of shapes, as this can affect how you approach the solution.

Learn how to calculate the perimeter of a rectangular patio constructed using 12 paving stones, each 6 feet long. Includes step-by-step solution and evaluation of mathematical expressions. Suitable for students in Grades 5-7.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation