Algebra Word Problem?
An open box is formed from a square piece of cardboard, by removing squares of side 6 in. from each corner and folding up the sides. If the volume of the carton is then 48 in^3, what was the length of a side of the original square of cardboard?
Answer
A. 12+4√5in.
B. 12+2√2in.
C. 6+2√2in.
D. 6+4√5in.
Comments
volume of new box = l × b × h
as it is a square
volume = l × l × h
l² = v/h = 48/6 = 8
l = √8 = 2√2
original length = 6 + 2√2 + 6 (6' from both sides added)
= 12 + 2√2
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Lets say that each side of the original square is x. When we cut away a 6 inch square from each corner, we make a kind of plus sign shape with a smaller square in the center. It's sides will be x - 12 units long. (Remember we cut away 6 inches on BOTH sides.)
When we fold up the sides of the plus sign to make a box, the smaller square will become the bottom of the box. The 6 inch edges will become the height of the box.
The volume of a box is found by multiplying the area of the base by the height of the box. The area of the base will be (x-12)(x-12). This equals (x-12)^2. When we multiply by the height we get 6(x-12)^2. According to the given information this must equal 48
6(x-12)^2 = 48 divide both sides by 6
(x-12)^2 = 8 take the square root of both sides
x-12 = 2 (sqrt 2) add 12 to both sides
x = 2(sqrt 2) + 12 which makes the answer B
x = original length
x-12 = length and width of base
6 = height
6(x-12)(x-12) = 48
x^2 - 24x + 144 = 8
x^2 - 24x +136 = 0
x = (24 +- 4â2)/2 = 12 +- 2â2
B