A rectangular box is to be made from a piece of cardboard 6 cm wide and 14 cm long by cutting out squares of the same size from the four corners and turning up the sides. If the volume of the box is to be 40 cm^3, what should the length of the side of the square to be cut out be?
base of box will be 6-2x by 14-2x , where 0 < x < 3
x(6-2x)(14-2x) = 40
x(2)(3-x)(2)(7-x) = 40
x(3-x)(7-x) = 10
x^3 - 10x^2 + 21x - 10 = 0
tried x = ±1, ±2, ±3
x = 2 worked
by synthetic division,
(x-2)(x^2 - 8x + 5) = 0
x = 2, the other roots are outside our restriction
the cut-out square should be 2 cm by 2 cm
Well, well, well, looks like we have a geometry problem in our hands. Don't worry, I'm here to make it fun!
Let's find out the length of the side of the square that needs to be cut out. First things first, we need to figure out the dimensions of the box when it's folded up.
When we cut out squares of the same size from all four corners and fold up the sides, the width of the box will be 6cm - 2x (square side length) and the length will be 14cm - 2x (square side length). Got it?
The volume of a rectangular box is given by the formula V = length × width × height. In this case, we know the volume is 40 cm³.
So, plugging in our dimensions, we have:
40 cm³ = (14 cm - 2x) × (6 cm - 2x) × x.
Now, let's simplify!
40 cm³ = (84 cm² - 28 cmx - 12 cmx + 4x²) × x.
40 cm³ = 4x³ - 40x² + 84x.
Okay, let's rearrange this equation for a more manageable solution.
4x³ - 40x² + 84x - 40 cm³ = 0.
Now, my dear friend, it's time to put our trust in the wonderful world of mathematical calculations to find the value of x, the side length of our square.
Once you've found that value, you'll know exactly what side length will give us a volume of 40 cm³.
Good luck with your calculations! If you need more help, let me know. I'm here to bring a little chuckle to your math journey.
To find the length of the side of the square to be cut out, we can use the formula for the volume of a rectangular box:
Volume = length × width × height
Given:
Width = 6 cm
Length = 14 cm
Volume = 40 cm^3
Since squares are cut out from the four corners of the cardboard, the width and length of the cardboard are reduced by twice the size of the cut-out square.
Let's denote the length of the side of the square to be cut out as "x".
The length of the box will be the original length of the cardboard minus twice the length of the cut-out squares:
Length of the box = 14 cm - 2x
Similarly, the width of the box will be the original width of the cardboard minus twice the length of the cut-out squares:
Width of the box = 6 cm - 2x
Now, we can substitute these values into the volume formula:
Volume = (14 - 2x) × (6 - 2x) × x
Expanding the equation:
40 = (84 - 28x - 12x + 4x^2) × x
40 = (4x^2 - 40x + 84) × x
Simplifying:
40 = 4x^3 - 40x^2 + 84x
Rearranging the equation:
4x^3 - 40x^2 + 84x - 40 = 0
Now, we can solve this equation to find the value of "x" using any appropriate method such as factoring, using the rational root theorem, or using numerical methods.
To find the length of the side of the square to be cut out, we can follow these steps:
1. Visualize the problem: Imagine a rectangular piece of cardboard with length (L) 14 cm, and width (W) 6 cm.
2. Identify the variables: Let x represent the length of the side of the square to be cut out from each corner.
3. Calculate the dimensions of the box: If squares of side length x are cut out from each corner, the dimensions of the resulting box will be:
- Length: L - 2x
- Width: W - 2x
- Height: x
4. Calculate the volume of the box: The volume of a rectangular box is calculated by multiplying the length, width, and height. In this case, the volume is given as 40 cm^3:
(L - 2x) * (W - 2x) * x = 40
5. Simplify and solve the equation: Substitute the given values into the equation and solve for x:
(14 - 2x) * (6 - 2x) * x = 40
Let's simplify this equation further by multiplying out the terms:
(84 - 28x - 12x + 4x^2) * x = 40
Combine like terms and multiply through by x:
(4x^2 - 40x + 84) * x = 40x
Simplify the equation:
4x^3 - 40x^2 + 84x - 40x = 0
Combine like terms again:
4x^3 - 40x^2 + 44x = 0
Factor out the common factor (4x):
4x(x^2 - 10x + 11) = 0
Apply the zero-product property and set each factor equal to zero:
4x = 0 or x^2 - 10x + 11 = 0
Solve the first equation to find x:
x = 0
Solve the second equation by factoring or using the quadratic formula. In this case, the quadratic equation does not factor, so we will use the quadratic formula:
x = ( -(-10) ± √((-10)^2 - 4(1)(11)) ) / (2(1))
Simplify the equation:
x = (10 ± √(100 - 44)) / 2
x = (10 ± √56) / 2
x = (10 ± 2√14) / 2
x = 5 ± √14
Since the length of a side cannot be negative, we discard the negative solution:
x = 5 + √14
Therefore, the length of the side of the square to be cut out should be 5 + √14 cm.