Answer:
![\bigg(\frac{2}{3} {y} \bigg)^{2}](https://tex.z-dn.net/?f=%5Cbigg%28%5Cfrac%7B2%7D%7B3%7D%20%20%7By%7D%20%5Cbigg%29%5E%7B2%7D)
STEP BY STEP EXPLANATION
![\frac{1}{4} {x}^{2} - \bigg( \frac{2}{3} \bigg)xy \\ \\ = \bigg(\frac{1}{2} {x} \bigg)^{2} - 2. \bigg(\frac{1}{2} {x} \bigg)\bigg( \frac{2}{3} y \bigg ) + \bigg(\frac{2}{3} {y} \bigg)^{2} \\ \\ = \bigg(\frac{1}{2} {x} - \frac{2}{3} y\bigg)^{2} \\](https://tex.z-dn.net/?f=%20%5Cfrac%7B1%7D%7B4%7D%20%20%7Bx%7D%5E%7B2%7D%20%20-%20%20%5Cbigg%28%20%5Cfrac%7B2%7D%7B3%7D%20%20%5Cbigg%29xy%20%5C%5C%20%20%5C%5C%20%20%3D%20%20%20%5Cbigg%28%5Cfrac%7B1%7D%7B2%7D%20%20%7Bx%7D%20%5Cbigg%29%5E%7B2%7D%20%20-%202.%20%5Cbigg%28%5Cfrac%7B1%7D%7B2%7D%20%20%7Bx%7D%20%5Cbigg%29%5Cbigg%28%20%5Cfrac%7B2%7D%7B3%7D%20y%20%5Cbigg%20%29%20%2B%20%20%5Cbigg%28%5Cfrac%7B2%7D%7B3%7D%20%20%7By%7D%20%5Cbigg%29%5E%7B2%7D%20%5C%5C%20%20%5C%5C%20%20%3D%20%5Cbigg%28%5Cfrac%7B1%7D%7B2%7D%20%20%7Bx%7D%20%20-%20%20%5Cfrac%7B2%7D%7B3%7D%20y%5Cbigg%29%5E%7B2%7D%20%20%5C%5C%20)
To make
a perfect square we should add ![\purple{\bold{\bigg(\frac{2}{3} {y} \bigg)^{2}}}](https://tex.z-dn.net/?f=%5Cpurple%7B%5Cbold%7B%5Cbigg%28%5Cfrac%7B2%7D%7B3%7D%20%20%7By%7D%20%5Cbigg%29%5E%7B2%7D%7D%7D)
The correct answer is the first one
The dimensions on the blueprint of the living room is 3 in by 4 in
We don’t know the value of the shorter side, so we will categorize it as x. Side 2 is just 4 feet longer than x, so we would add 4 on to it. Side 3 has double the x, so we would multiply it be 2 for 2x, and subtract the 4 feet from it.
Side 1: x
Side 2: x + 4
Side 3: 2x - 4
If the perimeter is 64 feet, then all of the sides have to add up to it. Therefore, first we add all of the side lengths up:
x + x + 4 + 2x - 4 = 4x.
Now we put 4x, the amount of all these sides added up, equal to the perimeter of 64.
4x = 64. Divide both sides by 4 to get x by itself.
x = 16.
Now that we know x is 16, we will substitute it in for all the side lengths’ equations.
We know that Side 1 was just x, so that will be 16. Since Side 2 was 4 more than x, we’d do 16 + 4 = 20. We substitute 16 in for x in Side 3’s equation: 2(16) - 4 = 32 - 4 = 28.
Therefore, the final lengths of all the sides are:
Side 1: 16
Side 2: 20
Side 3: 28
448 square inches.
The surface area of the box is 448 square inches so that is the least amount of paper needed to cover the box.