Math  /  Geometry

QuestionA box with a square base and no top is to be made from a square piece of carboard by cutting 3 in. squares from each corner and folding up the sides. The box is to hold 1587in31587 \mathrm{in}^{3}. How big a piece of cardboard is needed?
Your answer is: \square in. by \square in.

Studdy Solution

STEP 1

What is this asking? We need to find the size of a square piece of cardboard needed to make an open-top box with a square base that holds a specific volume, after cutting squares from the corners. Watch out! Don't forget that the original cardboard size changes after we cut out the squares!

STEP 2

1. Set up the equation
2. Solve for the side

STEP 3

Imagine a square piece of cardboard.
Now, picture cutting out a $3\$3 inch by $3\$3 inch square from *each* corner.
When we fold up the sides, we get a box!

STEP 4

Let xx be the original side length of the square piece of cardboard in inches.

STEP 5

After cutting out the $3\$3 inch squares, the base of the box will have side length x23=x6x - 2 \cdot 3 = x - 6 inches.
The height of the box will be $3\$3 inches.

STEP 6

The volume of the box is given by length times width times height.
Since the base is a square, the volume is (x6)(x6)3(x - 6) \cdot (x - 6) \cdot 3 cubic inches.
We're told the box needs to hold **15871587 cubic inches**, so we can set up an equation: (x6)23=1587(x - 6)^2 \cdot 3 = 1587.

STEP 7

Let's make this equation easier to work with!
Divide both sides by **33**: (x6)2=15873=529(x - 6)^2 = \frac{1587}{3} = 529.

STEP 8

To get rid of the square, we take the square root of both sides: x6=±529=±23x - 6 = \pm \sqrt{529} = \pm 23.

STEP 9

This gives us two possible solutions: x6=23x - 6 = 23 or x6=23x - 6 = -23.
Solving the first one, we add 66 to both sides to get x=23+6=29x = 23 + 6 = 29.
Solving the second one, we add 66 to both sides to get x=23+6=17x = -23 + 6 = -17.

STEP 10

Since we're dealing with a length, we can't have a negative value!
So, we discard x=17x = -17.
Our original cardboard needs to be x=29x = \textbf{29} inches per side.

STEP 11

We need a 29\textbf{29} in. by 29\textbf{29} in. piece of cardboard.

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