A box with no top is to be constructed from a piece of cardboard whose width measures x cm and whose length measures 6 cm more than its width. The box is to be formed by cutting squares that measure 2 cm on each side from the four corners, and then folding up the sides. If the volume of the box will be 182cm^3,what are the dimensions of the piece of cardboard?
To find the dimensions of the cardboard, we need to set up equations based on the given information.
Let's assume the width of the cardboard is x cm. According to the problem, the length of the cardboard is 6 cm more than its width. So the length would be x + 6 cm.
To construct the box, we need to cut squares that measure 2 cm on each side from the four corners and then fold up the sides. If we cut 2 cm from each corner, the width and length of the base of the box will be reduced by 4 cm (2 cm from each side).
So the width of the base of the box after cutting would be x - 4 cm, and the length would be (x + 6) - 4 cm.
Now, let's calculate the volume of the box. The volume of a rectangular prism (box) is given by the formula:
Volume = length * width * height
In this case, the box has no top, so the height would be the length of the squares we cut from the corners, which is 2 cm.
Given that the volume of the box is 182 cm^3, we can set up the equation:
182 = (x - 4) * ((x + 6) - 4) * 2
Now, let's solve this equation to find the value of x, which will give us the width of the cardboard.
182 = (x - 4) * (x + 2) * 2
91 = (x - 4) * (x + 2)
91 = x^2 - 2x - 8
0 = x^2 - 2x - 99
To solve this quadratic equation, we can use factoring, completing the square method, or the quadratic formula. In this case, the quadratic equation is not easily factorable, so let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 1, b = -2, and c = -99. Plugging these values into the quadratic formula, we get:
x = (2 ± √((-2)^2 - 4 * 1 * (-99))) / (2 * 1)
x = (2 ± √(4 + 396)) / 2
x = (2 ± √400) / 2
x = (2 ± 20) / 2
Now, let's solve for both values of x:
Case 1: x = (2 + 20) / 2
x = 22 / 2
x = 11 cm
Case 2: x = (2 - 20) / 2
x = -18 / 2
x = -9 cm
Since the width cannot be negative, we discard the second case.
Therefore, the width of the cardboard is 11 cm.
To find the length of the cardboard, we can substitute this value back into the equation x + 6. In this case, the length would be:
Length = 11 + 6 = 17 cm
So, the dimensions of the piece of cardboard are width = 11 cm and length = 17 cm.
Let's break down the problem step-by-step:
Step 1:
Let's assume the width of the cardboard is "x" cm.
According to the problem, the length of the cardboard is 6 cm more than its width.
Therefore, the length of the cardboard is (x + 6) cm.
Step 2:
To construct the box, we need to cut squares with sides measuring 2 cm from each corner.
This means that the width of the bottom of the box will be reduced by 4 cm (2 cm from each side).
Similarly, the length of the bottom of the box will be reduced by 4 cm as well.
Step 3:
After cutting the squares from each corner, the dimensions of the bottom of the box will be:
Width = (x - 4) cm
Length = ((x + 6) - 4) cm
Step 4:
To determine the height of the box, we need to fold up the sides.
The height of the box will be equal to the side of the squares cut from the corners, which is 2 cm.
Step 5:
Now, we can calculate the volume of the box using the given information.
Volume = Length × Width × Height
Given: Volume = 182 cm^3
Volume = ((x - 4) cm) × (((x + 6) - 4) cm) × 2 cm
Step 6:
Now, set up the equation and solve for x:
182 cm^3 = (x - 4) cm × (x + 2) cm × 2 cm
Step 7:
Simplify the equation:
182 cm^3 = (2x - 4) cm × (x + 2) cm
Step 8:
Expand and rearrange the equation:
182 cm^3 = (2x^2 + 4x - 4x - 8) cm^2
Step 9:
Simplify further:
182 cm^3 = (2x^2 - 8) cm^2
Step 10:
Divide both sides by 2:
91 cm^3 = x^2 - 4 cm^2
Step 11:
Rearrange the equation, bringing everything to one side:
x^2 - 4 cm^2 - 91 cm^3 = 0
Step 12:
Factor the equation:
(x + 7 cm) × (x - 13 cm) = 0
Step 13:
Solve for x:
x + 7 cm = 0 or x - 13 cm = 0
Step 14:
If x + 7 cm = 0, then x = -7 cm. Since we're dealing with measurements, it wouldn't make sense for the width to be negative. Therefore, we discard this solution.
Step 15:
If x - 13 cm = 0, then x = 13 cm.
Step 16:
Therefore, the width of the cardboard is x = 13 cm.
Step 17:
The length of the cardboard is 6 cm more than its width, so the length is x + 6 cm = 13 cm + 6 cm = 19 cm.
Step 18:
To summarize, the dimensions of the piece of cardboard are:
Width = 13 cm
Length = 19 cm