An open 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 box is to be 144 cubic centimeters. Find the size of the original piece of material.
If the side is x cm, then
4(x-2*4)^2 = 144
Or, since the height of the box is 4 cm, and 144/4 = 36, the final base of the box is 6x6 cm.
That makes the original size 14x14
Well, well, an open box puzzle! Let's see if I can help you solve it with a touch of humor.
Since we're dealing with squares, let's call the length of one side of the original square piece of material "x" centimeters. After we cut out those four-corner squares, the length and width of the base of the box will be (x-8) cm (because we cut 4 cm from each corner).
Now, if we turn up the sides, we get a box with a height of 4 cm. So, the volume of the box is given by length times width times height.
Using this information, we can set up the equation: (x-8)(x-8)(4) = 144.
Simplifying, we have (x-8)(x-8) = 36.
Now expand that equation: x^2 - 16x + 64 = 36.
Rearranging, we have: x^2 - 16x + 28 = 0.
Now, my friend, use the quadratic formula and you'll find that x = 14 (or x = 2, but that doesn't make much sense in this context).
So the size of the original piece of material is 14 centimeters. Voila!
Hope my mathematical humor entertained you along the way!
To find the size of the original piece of material, we need to determine the dimensions of the open box.
Let's assume that the original square piece of material has side length 'x' centimeters.
When we cut four-centimeter squares from each corner, the resulting dimensions of the base of the box will be (x - 8) centimeters by (x - 8) centimeters.
Since we turn up the sides, the height of the box will be 4 cm.
The volume of a rectangular prism is given by the formula V = length × width × height.
Given that the volume of the finished box is 144 cubic centimeters, we can set up the equation as follows:
144 = (x - 8) × (x - 8) × 4
Let's solve this equation step-by-step to find the value of 'x':
1. Simplifying the equation:
144 = 4(x - 8)²
2. Dividing both sides by 4:
36 = (x - 8)²
3. Taking the square root of both sides:
√(36) = (√(x - 8)²)
4. Simplifying the equation:
6 = x - 8
5. Adding 8 to both sides:
6 + 8 = x - 8 + 8
6. Simplifying the equation:
14 = x
Therefore, the original square piece of material has a side length of 14 centimeters.
To find the size of the original piece of material, we can break down the problem into smaller steps:
Step 1: Determine the dimensions of the box.
Let's assume that the side length of the square piece of material is 'x' centimeters.
When you cut four 4-centimeter squares from each corner, the resulting length and width of the base of the box will be 'x - 8' centimeters (since you remove 4 cm from each side). The height of the box will be 4 cm.
Step 2: Calculate the volume of the box.
The volume of a rectangular box is given by the formula: Volume = Length × Width × Height.
In this case, the length is 'x - 8', the width is 'x - 8', and the height is 4 cm.
So, the volume is: Volume = (x - 8) × (x - 8) × 4.
Step 3: Set up the equation.
Since the volume of the finished box is 144 cubic centimeters, we can set up the equation: (x - 8) × (x - 8) × 4 = 144.
Step 4: Solve the equation.
Let's solve the equation for x:
(x - 8) × (x - 8) × 4 = 144.
Expand the equation: 4(x² - 16x + 64) = 144.
Distribute 4: 4x² - 64x + 256 = 144.
Move 144 to the other side: 4x² - 64x + 256 - 144 = 0.
Simplify the equation: 4x² - 64x + 112 = 0.
Divide the equation by 4: x² - 16x + 28 = 0.
Now, we can factorize or use the quadratic formula to solve for x. Factoring may not be possible. So, let's use the quadratic formula:
x = (-b ± sqrt(b² - 4ac)) / 2a.
In this equation, a = 1, b = -16, and c = 28.
x = (-(-16) ± sqrt((-16)² - 4(1)(28))) / 2(1).
x = (16 ± sqrt(256 - 112)) / 2.
x = (16 ± sqrt(144)) / 2.
x = (16 ± 12) / 2.
Solving for the positive and negative values:
x₁ = (16 + 12) / 2 = 28 / 2 = 14.
x₂ = (16 - 12) / 2 = 4 / 2 = 2.
Since the original piece of material cannot have a side length of 2 cm (since we remove 4 cm squares from each corner), the size of the original piece of material is 14 centimeters.