a rectangular piece of cardboard is twice as long as it is wide . from each of its for corners, a square piece 3 inches on a side cut out. the flaps at each corner are then turned up to form an open box. if the volume of the box is 168 cubic inches, what were the original dimensions of the piece of cardboard?
Well, well, we have quite the geometry puzzle here! Let's break it down and find the dimensions of that mysterious piece of cardboard.
Let's start by assuming that the width of the cardboard is "w" inches. Since the length is twice as long as the width, that makes it 2w inches.
Now, let's picture the situation. We have four squares, each with sides measuring 3 inches, cut out from the corners. After folding the flaps up, it forms an open box. The length of this box is w - 2 times the side length of the square (which is 3 inches). So, the length of the box becomes 2w - 6 inches. The width of the box, similarly, becomes w - 6 inches.
The height of the box is simply 3 inches since that's the remaining portion of the cardboard after the squares are cut out.
Now, the volume of the box is given as 168 cubic inches. Volume, as we all know, is calculated by multiplying length × width × height. So, we can set up the equation:
(w - 6)(2w - 6)(3) = 168
Now we solve it:
6w^2 - 36w + 36w - 216 = 168
6w^2 - 216 = 168
6w^2 = 384
w^2 = 64
w = 8
So, the original dimensions of the piece of cardboard were 8 inches by 16 inches! Voilà!
Let's go step-by-step to solve this problem:
Step 1: Assign variables
Let's assign a variable to represent the width of the rectangular piece of cardboard. Let's call it "w". Since the length is twice the width, we can assign the length as "2w".
Step 2: Calculate the dimensions after cutting the squares
We need to subtract the length of the squares cut out from each side of the rectangle. Since the squares are 3 inches on each side, the new dimensions (length and width) would be reduced by 3 inches on each side.
So, the new length would be "2w - 2(3)" and the new width would be "w - 2(3)".
Step 3: Calculate the volume of the resulting box
To calculate the volume of the box, we multiply the new length, new width, and the width of the square cutouts. The volume of the box is given as 168 cubic inches, so we have:
(2w - 2(3))(w - 2(3))(3) = 168
Step 4: Simplify the equation and solve for w
By simplifying the equation, we get:
(2w - 6)(w - 6)(3) = 168
(2w - 6)(w - 6) = 56
Step 5: Expand and solve the quadratic equation
Expanding the equation, we get:
2w^2 - 18w + 36 = 56
2w^2 - 18w - 20 = 0
Step 6: Factor and solve the quadratic equation
Factoring the quadratic equation, we get:
(2w + 2)(w - 10) = 0
Setting each factor equal to zero, we have two possible solutions:
2w + 2 = 0 or w - 10 = 0
Solving each equation, we get:
2w = -2 or w = 10
Since the width cannot be negative, we disregard the first equation.
Step 7: Calculate the original dimensions
We know that the width is 10 inches. To find the length, we substitute the value of w into the equation for the length:
Length = 2w = 2(10) = 20 inches
Therefore, the original dimensions of the piece of cardboard were 20 inches by 10 inches.
To determine the original dimensions of the piece of cardboard, we need to use the given information and solve step by step.
Let's start by assigning a variable to one of the unknown dimensions. Let's say that the width of the cardboard is "x" inches.
According to the problem, the length of the cardboard is twice its width. Therefore, the length would be 2x inches.
Now, let's visualize the situation. If you cut out a 3x3 square from each corner, you would be left with four flaps that can be folded up to create the sides of the box.
The height of the box is determined by the side length of the square cut out, which is 3 inches. Therefore, the height would be 3 inches.
The remaining dimensions of the cardboard after the corners are cut out would be (2x - 3) inches for the length, (x - 3) inches for the width, and 3 inches for the height.
To find the volume of the box, we multiply the length, width, and height. According to the problem, the volume is given as 168 cubic inches.
So we have the equation: (2x - 3) * (x - 3) * 3 = 168.
Now, we can solve this equation for x.
Expanding the equation: 6x^2 - 27x + 27 = 168.
Rearranging to set the equation to zero: 6x^2 - 27x + 27 - 168 = 0.
Combining like terms: 6x^2 - 27x - 141 = 0.
Now we have a quadratic equation. To solve it, we can factor or use the quadratic formula.
To make things simpler, let’s use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a.
For our equation, a = 6, b = -27, and c = -141.
Plugging the values into the quadratic formula: x = (-(-27) ± √((-27)^2 - 4 * 6 * -141)) / (2 * 6).
Simplifying further: x = (27 ± √(729 + 3384)) / 12.
Calculating within the square root: x = (27 ± √(4113)) / 12.
The values for x will be both positive and negative, but since we are dealing with dimensions, we only consider the positive value.
Calculating the square root (approximated to three decimal places): x ≈ 6.617.
Therefore, the width of the original cardboard is approximately 6.617 inches.
Since the length is twice the width, the length would be approximately 2 * 6.617 = 13.234 inches.
Hence, the original dimensions of the piece of cardboard were approximately 6.617 inches by 13.234 inches.
Length (L) = 2x
Width (W) = x
Height (H) =3
Volume (V) =168
Formula for the volume:
V=L x W x H
Hence,
(2x)(x)(3)=168
6x^2=168
x^2=168/6
x= 2 times the sqrt of 7
Answer:
W= 2 times the sqrt of 7 inches
L= 4 times the sqrt of 7 inches