- Explain her error, and include the correct expression for the perimeter. The width is (5/6x + 23) and the height is 12.
Michelle simplified an expression for the perimeter of the rectangle below and got (5/6x + 23) + (5/6x + 23) + 12. Explain her error, and include the correct expression for the perimeter. The width is (5/6x + 23) and the height is 12.
The Correct Answer and Explanation is:
Michelle made an error in simplifying the expression for the perimeter of the rectangle.
The formula for the perimeter of a rectangle is: P=2×width+2×heightP = 2 \times \text{width} + 2 \times \text{height}P=2×width+2×height
In this case, the width is given as (56x+23)\left( \frac{5}{6}x + 23 \right)(65x+23) and the height is given as 12. Therefore, the correct expression for the perimeter should be: P=2×(56x+23)+2×12P = 2 \times \left( \frac{5}{6}x + 23 \right) + 2 \times 12P=2×(65x+23)+2×12
Now, let’s break it down:
- Multiply the width by 2:
2×(56x+23)=106x+46=53x+462 \times \left( \frac{5}{6}x + 23 \right) = \frac{10}{6}x + 46 = \frac{5}{3}x + 462×(65x+23)=610x+46=35x+46
- Multiply the height by 2:
2×12=242 \times 12 = 242×12=24
So, the correct expression for the perimeter is: P=53x+46+24P = \frac{5}{3}x + 46 + 24P=35x+46+24
- Combine the constant terms:
P=53x+70P = \frac{5}{3}x + 70P=35x+70
Explanation of Michelle’s error:
Michelle mistakenly simplified the perimeter expression as: (56x+23)+(56x+23)+12\left( \frac{5}{6}x + 23 \right) + \left( \frac{5}{6}x + 23 \right) + 12(65x+23)+(65x+23)+12
This is incorrect because she did not account for the full perimeter formula. Instead of multiplying the width by 2 and the height by 2, she added the width twice and added the height only once. The correct approach requires multiplying both the width and height by 2 before summing them to find the perimeter.
Thus, the correct perimeter expression is: P=53x+70P = \frac{5}{3}x + 70P=35x+70
