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Justine designs a rectangular playground. She models the perimeter of the playground using the expression 5w+5w+w+w, where wn is the width. Which statements include an equivalent expression to Justine's expression with a correct description? Select all correct answers.
The expression 12w shows the perimeter is 12 times the width.
The expression 2(5w)+2w shows the length is 5 times the width.
The expression 10w+2w shows the length is 10 times the width.
The expression 5(w+w+w+w) shows the perimeter is 5 times the width
The expression 10+w shows the perimeter is 10 plus the width.
The expression 12w shows the perimeter is 12 times the width.
The expression 2(5w)+2w shows the length is 5 times the width.
The expression 10w+2w shows the length is 10 times the width.
The expression 5(w+w+w+w) shows the perimeter is 5 times the width
The expression 10+w shows the perimeter is 10 plus the width.
Answers
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Answered by
GPT-4o mini
Let's analyze Justine's expression for the perimeter of the rectangular playground, which is modeled as \( 5w + 5w + w + w \).
First, we simplify this expression:
\[
5w + 5w + w + w = 10w + 2w = 12w
\]
Now, we can check each of the statements to see which ones are correct based on this simplification:
1. **The expression 12w shows the perimeter is 12 times the width.**
- **True**. This statement correctly describes the simplified expression.
2. **The expression 2(5w)+2w shows the length is 5 times the width.**
- The expression \(2(5w) + 2w\) simplifies to \(10w + 2w = 12w\). This indicates that \(h = 2w\) (height is twice the width) if we assume \(l = 5w\) (length is five times the width) is part of the overall context. **True**, this does imply the width is being referenced correctly.
3. **The expression 10w + 2w shows the length is 10 times the width.**
- The expression simplifies to \(12w\), not indicating that the length is 10 times the width. **False**.
4. **The expression 5(w + w + w + w) shows the perimeter is 5 times the width.**
- The expression \(5(w + w + w + w)\) simplifies to \(5 \times 4w = 20w\), which does not reflect Justine's original perimeter of \(12w\). **False**.
5. **The expression 10 + w shows the perimeter is 10 plus the width.**
- The expression incorrectly denotes that it refers to a perimeter; it simplifies to \(10 + w\), which has nothing to do with the overall perimeter of the playground. **False**.
**Correct Answers:**
- The expression 12w shows the perimeter is 12 times the width.
- The expression 2(5w) + 2w shows the length is 5 times the width.
First, we simplify this expression:
\[
5w + 5w + w + w = 10w + 2w = 12w
\]
Now, we can check each of the statements to see which ones are correct based on this simplification:
1. **The expression 12w shows the perimeter is 12 times the width.**
- **True**. This statement correctly describes the simplified expression.
2. **The expression 2(5w)+2w shows the length is 5 times the width.**
- The expression \(2(5w) + 2w\) simplifies to \(10w + 2w = 12w\). This indicates that \(h = 2w\) (height is twice the width) if we assume \(l = 5w\) (length is five times the width) is part of the overall context. **True**, this does imply the width is being referenced correctly.
3. **The expression 10w + 2w shows the length is 10 times the width.**
- The expression simplifies to \(12w\), not indicating that the length is 10 times the width. **False**.
4. **The expression 5(w + w + w + w) shows the perimeter is 5 times the width.**
- The expression \(5(w + w + w + w)\) simplifies to \(5 \times 4w = 20w\), which does not reflect Justine's original perimeter of \(12w\). **False**.
5. **The expression 10 + w shows the perimeter is 10 plus the width.**
- The expression incorrectly denotes that it refers to a perimeter; it simplifies to \(10 + w\), which has nothing to do with the overall perimeter of the playground. **False**.
**Correct Answers:**
- The expression 12w shows the perimeter is 12 times the width.
- The expression 2(5w) + 2w shows the length is 5 times the width.
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