Let’s walk through the steps to complete each part of the problem regarding Valerie's and Seth's measurements of the house's perimeter.
Part A
For Valerie's measurements:
- 4 walls at 2.5 feet each: \( 4 \times 2.5 \)
- 3 walls at 21.25 feet each: \( 3 \times 21.25 \)
- 1 wall at 32 feet: \( 32 \)
So, the expression for the total length of the walls that Valerie measured is: \[ (2.5 \times 4) + (21.25 \times 3) + 32 \]
Part B
For Seth's measurements:
- 1 wall at 3.5 feet: \( 3.5 \)
- 2 walls at 22.75 feet each: \( 2 \times 22.75 \)
- 1 wall at 58 feet: \( 58 \)
So, the expression for the total length of the walls that Seth measured is: \[ 3.5 + (22.75 \times 2) + 58 \]
Part C
To find the expression for the perimeter of the house, we add Valerie's and Seth's expressions together: \[ (2.5 \times 4) + (21.25 \times 3) + 32 + 3.5 + (22.75 \times 2) + 58 \]
Part D
Now, we simplify the expression from part C step-by-step:
- Calculate each component:
- \( 2.5 \times 4 = 10 \)
- \( 21.25 \times 3 = 63.75 \)
- \( (22.75 \times 2) = 45.5 \)
Now, substitute these values into the expression: \[ 10 + 63.75 + 32 + 3.5 + 45.5 + 58 \]
Add them:
- \( 10 + 63.75 = 73.75 \)
- \( 73.75 + 32 = 105.75 \)
- \( 105.75 + 3.5 = 109.25 \)
- \( 109.25 + 45.5 = 154.75 \)
- \( 154.75 + 58 = 212.75 \)
So, the total perimeter of the house is: \[ \text{Perimeter} = 212.75 \text{ feet} \]
Part E
If you evaluate Valerie’s walls and Seth’s walls separately and then add the two expressions together, you will indeed get the same answer as in part D. Here is the breakdown of the calculations:
-
First, evaluate Valerie’s expression:
- \( 10 + 63.75 + 32 = 105.75 \)
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Next, evaluate Seth’s expression:
- \( 3.5 + 45.5 + 58 = 107 \)
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Now, add these two results together:
- \( 105.75 + 107 = 212.75 \)
Thus, adding the individual results gives the same total perimeter: \[ \text{Perimeter} = 212.75 \text{ feet} \]
The answer is the same because addition is associative and commutative, meaning that it does not matter in which order you add the lengths; you will end up with the same total perimeter.
Self-Evaluation
On a scale of 1 to 5, I would rate my work a 5. I clearly understood how to break down the measurements from both Valerie and Seth systematically and combined them accurately. The challenge was in ensuring that I stayed organized while maintaining accuracy in the calculations. I learned the importance of stepwise evaluation to avoid confusion in multi-part problems like this.