An open gift box is to be made from a square piece of material by cutting four-centimeter squares from each corner and turning up the sides (see figure). The volume of the finished gift box is to be 324 cubic centimeters. Find the size of the original piece of material.
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To find the size of the original piece of material, we need to determine the dimensions of the square and the height of the box.
Let's start by visualizing the problem. We have a square piece of material, and we cut four squares with a side length of 4 centimeters from each corner. We then fold up the sides to create a box.
Let's assign variables to the dimensions:
Let x be the side length of the original square piece of material.
Let h be the height of the box.
Since we cut out squares with a side length of 4 centimeters from each corner, the length of each side of the bottom of the box (after folding) will be x - 8 centimeters. The height of the box will be 4 centimeters.
The volume of the box can be calculated by multiplying the length, width, and height:
Volume = Length × Width × Height
In this case, the length and width of the bottom of the box will be the same (x - 8), and the height is given as 4 centimeters. So we have the equation:
Volume = (x - 8) × (x - 8) × 4
The problem states that the volume of the finished gift box is 324 cubic centimeters. So we can set up the equation:
324 = (x - 8) × (x - 8) × 4
Now, let's solve this equation for x:
324 = 4 × (x - 8)²