A rectanguler piece of metal is 5 inches longer than it is wide. Square with sides 1 inches longer are cut from the four corners and the flaps are folded upward to form an open box. If the volume of the box is 644 inches, what are the original dimensions of the piece of metal?
original dimensions: x and x+5
(x-2)(x+5-2)(1) = 644
Now just solve for x.
25W
45L
Why was the rectangular piece of metal feeling so boxy? It was longing for some scissors to cut it into shape!
Let's solve this conundrum. Let's say the width of the rectangular piece is x inches. According to the problem, the length of the piece is 5 inches longer than its width, which makes it (x + 5) inches long.
Now, we cut squares from the corners with sides measuring 1 inch. When we fold the flaps upward, it forms an open box. The dimensions of the box's height, length, and width will be (x-2), (x+5-2), and (x-2), respectively.
To find the volume of the box, we multiply these dimensions:
(x - 2)(x + 3)(x - 2) = 644
Expanding this equation, we get:
(x^2 - 2x + 3x - 6)(x - 2) = 644
(x^2 + x - 6)(x - 2) = 644
x^3 + x^2 - 2x^2 - 2x - 6x + 12 = 644
x^3 - x^2 - 8x - 632 = 0
Now let me grab my imaginary calculator...
Beep boop beep boop... crunching numbers...
After some mathematical acrobatics, I can humorously deduce that the original dimensions of the piece of metal are approximately 15.1 inches long and 10.1 inches wide.
So, let's thank those scissors for bringing laughter and helping us discover the original dimensions of the rectangular piece of metal!
To solve this problem, we can break it down into smaller steps:
Step 1: Understand the problem.
We are given a rectangular piece of metal. It is 5 inches longer than it is wide. The four corners are cut to form a square with sides that are 1 inch longer, and the flaps are folded upward to create an open box. We are given that the volume of the box is 644 inches.
Step 2: Define the variables.
Let's define the width of the rectangular piece of metal as "w" inches.
Since the length is 5 inches longer than the width, we can define the length as "w + 5" inches.
Step 3: Find the dimensions of the box.
When the flaps are folded, the width and length will be decreased by twice the length of the square cut from each corner.
Since the square has sides that are 1 inch longer, the length and width of the resulting box will be "w - 2 × 1" and "w + 5 - 2 × 1" respectively.
Step 4: Calculate the volume of the box.
The volume of a rectangular box is given by the formula: volume = length × width × height.
We are given that the volume of the box is 644 cubic inches. So we can write the equation: 644 = (w - 2) × (w + 5 - 2) × 1.
Step 5: Solve the equation.
Simplifying the equation:
644 = (w - 2) × (w + 3)
Expanding the equation:
644 = w^2 + 3w - 2w - 6
Rearranging the terms:
w^2 + w - 638 = 0
Step 6: Solve the quadratic equation.
Solving the quadratic equation w^2 + w - 638 = 0 can be done by factoring, completing the square, or using the quadratic formula. In this case, let's use the quadratic formula:
w = [-b ± √(b^2 - 4ac)] / (2a)
Using the quadratic formula:
w = [-1 ± √(1^2 - 4 × 1 × -638)] / (2 × 1)
w = [-1 ± √(1 + 2552)] / 2
w = [-1 ± √(2553)] / 2
Step 7: Solve for the width.
Taking the positive square root:
w = (-1 + √2553) / 2
Calculating the width:
w ≈ 31.9 inches (rounded to one decimal place)
Step 8: Solve for the length.
Using the width value, we can find the length:
Length = width + 5
Length ≈ 31.9 + 5
Length ≈ 36.9 inches (rounded to one decimal place)
So, the original dimensions of the piece of metal were approximately 31.9 inches wide and 36.9 inches long.