A box with no top is to be constructed from a piece of cardboard whose length measures 6 inch more than its width. The box is to be formed by cutting squares that measure 2 inches on each side from the four corners an then folding up the sides. If the volume of the box will be 110 inches cubed what are the dimensions of the piece of cardboard?
Let x = width of cardboard
Therefore L = x+6
volume of open box is 110 in^3
cut squares are 2 inches = H
V=HXWxL=110
H=2
W=x-4
L=(x+6)-4
2(x-4)(x+2)=110 ==> divide both sides by 2
(x-4)(x+2)=55
x^2 -2x -8 = 55 ==> subtract 55 from both sides
X^2 -2x -63 = 0 ==> factor
(x-9)(x+7) = 0
x= 9 or -7 ==> x has to be greater than 0, therefore answer is 9
check
2((9)-4)((9)+6-4)=110
2(5)(11)=110
110=110 it checks
dimensions of cardboard is
W=9 inches
L=(9)+6=15 inches
Well, it seems like our cardboard box wants to play hide and seek, with no top and all. How mischievous!
Let's solve this puzzle together. First, let's figure out the dimensions of the cardboard before we start cutting it up like a fancy artist.
Let's assume that the width of the cardboard is "x" inches. Since the length is 6 inches more than the width, the length will be "x + 6" inches. So our cardboard has dimensions of x inches by (x + 6) inches.
Now, let's cut those little 2-inch squares from the corners. This reduces both the width and length of the cardboard box by 4 inches (2 inches from each side). After the squares are cut, the width becomes (x - 4) inches and the length becomes (x + 6 - 4) inches, which simplifies to (x + 2) inches.
To find the volume of the box, we need to multiply its dimensions, width, length, and height. Since the height of the box is the same as the length of the 2-inch square that we cut from each corner, the height is 2 inches.
Now, let's calculate the volume of the box using the formula: volume = length × width × height.
110 inches cubed = (x + 2) inches × (x - 4) inches × 2 inches
Phew! That was a lot of math, but don't worry, I'm here to help. Let's solve this equation:
2(x + 2)(x - 4) = 110
Simplifying further, we get:
2x^2 - 12x - 132 = 0
Now, we need to channel our inner mathematician and solve this quadratic equation. We could use the quadratic formula, but I have a better idea! Let's ask our imaginary friend, the Math Wizard, to help us out. Hey, Math Wizard, what are the solutions to this equation?
Math Wizard: The solutions are x = 12 and x = -5.5.
Hmm, something seems a little fishy with that negative value. Since the width of a cardboard can't be negative, we'll discard that solution. Therefore, the width of the cardboard is x = 12 inches.
Now, to find the length of the cardboard, we'll add 6 inches to the width: 12 inches + 6 inches = 18 inches.
Ta-da! The dimensions of the cardboard are 12 inches by 18 inches. We're all set to cut, fold, and create our wonderful box!
Remember, measuring twice and cutting once will save you from becoming a square peg in a round hole. Have fun building your box!
Let's assume the width of the cardboard is "x" inches.
According to the given information, the length of the cardboard is 6 inches more than its width, so the length would be "x + 6" inches.
To construct the box, we will remove 2-inch squares from each corner of the cardboard.
The height of the box would be 2 inches.
The length of the box would be (x + 6) - 2x = x + 6 - 2x = 6 - x inches.
The width of the box would be (x) - 2(2) = x - 4 inches.
The volume of the box is given as 110 cubic inches, so we can calculate the volume of the box as follows:
Volume = length * width * height
110 = (6 - x) * (x - 4) * 2
To solve this equation, we can multiply the factors on the right side:
110 = (6 - x) * (x - 4) * 2
110 = (6x - 24 - x^2 + 4x) * 2
110 = (10x - x^2 - 24) * 2
110 = 20x - 2x^2 - 48
Now, let's rearrange the equation to bring it to the form of a quadratic equation:
2x^2 - 20x + 158 = 0
To solve this quadratic equation, we can either factor it or use the quadratic formula. Since the equation does not easily factor, we will use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our case, a = 2, b = -20, and c = 158.
Plugging these values into the quadratic formula, we get:
x = (-(-20) ± √((-20)^2 - 4 * 2 * 158)) / (2 * 2)
x = (20 ± √(400 - 1264)) / 4
x = (20 ± √(-864)) / 4
Since we can't take the square root of a negative number in this context, it means that there is no real value for "x" that satisfies the given conditions.
To solve this problem, we can break it down into steps:
Step 1: Determine the dimensions of the box.
Let's assume the width of the piece of cardboard is "x" inches. According to the problem, the length is 6 inches more than the width, so it would be (x + 6) inches.
When we cut squares with sides measuring 2 inches from the corners, the resulting box will have a width and length reduced by 4 inches (2 inches on each side). Therefore, the dimensions of the box will be:
Width: (x - 4) inches
Length: [(x + 6) - 4] inches = (x + 2) inches
Step 2: Calculate the volume of the box.
The volume of a rectangular prism (box) is given by the formula: Volume = Length x Width x Height.
In this case, since the height is 2 inches (equal to the length of the cut squares), we have:
Volume = (x + 2) inches x (x - 4) inches x 2 inches
Volume = 2(x + 2)(x - 4) cubic inches
Step 3: Solve for x.
According to the problem, the volume of the box is 110 cubic inches. So, we can set up the equation:
110 = 2(x + 2)(x - 4)
To find the value of x, we can solve this equation.
Step 4: Solve the equation.
Begin by expanding and simplifying the equation:
110 = 2(x^2 - 2x - 8)
110 = 2x^2 - 4x - 16
Divide both sides of the equation by 2 to simplify:
55 = x^2 - 2x - 8
Rearrange the equation to form a quadratic equation:
x^2 - 2x - 63 = 0
Now, we need to solve this quadratic equation to find the value(s) of x.
Step 5: Solve the quadratic equation.
We can solve the quadratic equation by factoring, using the quadratic formula, or by completing the square.
Assuming we use factoring:
(x - 9)(x + 7) = 0
This equation is solved when either (x - 9) = 0 or (x + 7) = 0, which means:
x = 9 or x = -7
However, since the width and length cannot be negative in this context, we disregard the x = -7 solution.
Step 6: Find the dimensions of the piece of cardboard.
The width of the cardboard is x inches, so in this case, it would be:
Width = 9 inches
The length is 6 inches more than the width, so:
Length = 9 inches + 6 inches = 15 inches
Therefore, the dimensions of the piece of cardboard are:
Width = 9 inches
Length = 15 inches
That's it! The dimensions of the piece of cardboard are 9 inches by 15 inches.