A Rectangular Sheet Of Metal Has Identical Squares Cut From Each Corner. The Sheet Is Then Bent Along The Dotted Lines To Form An Open Box. The Volume Of The Box Is $420 , \text{in}^3$.The Equation $4x^3 - 72x^2 + 320x = 420$ Can

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Introduction

In this article, we will delve into a mathematical problem involving a rectangular sheet of metal with identical squares cut from each corner. The sheet is then bent along the dotted lines to form an open box. Our goal is to find the dimensions of the box, given that its volume is $420 , \text{in}^3$. To achieve this, we will use algebraic techniques to solve the equation $4x^3 - 72x^2 + 320x = 420$.

The Problem

We are given a rectangular sheet of metal with a length of $l$ inches, a width of $w$ inches, and a height of $h$ inches. The sheet has identical squares cut from each corner, with each side of the square having a length of $x$ inches. After cutting the squares, the sheet is bent along the dotted lines to form an open box.

The volume of the box is given by the formula $V = lwh$. Since the sheet is bent along the dotted lines, the length and width of the box are $l - 2x$ and $w - 2x$, respectively. The height of the box is $h$.

The Equation

We are given the equation $4x^3 - 72x^2 + 320x = 420$. This equation represents the volume of the box in terms of the side length of the cut squares, $x$.

To solve this equation, we can start by rearranging the terms to get $4x^3 - 72x^2 + 320x - 420 = 0$. This is a cubic equation in $x$, and we can use various algebraic techniques to solve it.

Solving the Equation

To solve the equation $4x^3 - 72x^2 + 320x - 420 = 0$, we can start by factoring out the greatest common factor, which is $4$. This gives us $4(x^3 - 18x^2 + 80x - 105) = 0$.

Next, we can try to factor the cubic expression inside the parentheses. After some trial and error, we find that $x^3 - 18x^2 + 80x - 105$ can be factored as $(x - 5)(x^2 - 13x + 21)$.

Factoring the Quadratic Expression

The quadratic expression $x^2 - 13x + 21$ can be factored as $(x - 7)(x - 3)$. Therefore, the entire equation can be written as $4(x - 5)(x - 7)(x - 3) = 0$.

Finding the Solutions

To find the solutions to the equation, we can set each factor equal to zero and solve for $x$. This gives us $x - 5 = 0$, $x - 7 = 0$, and $x - 3 = 0$.

Solving for $x$, we find that $x = 5$, $x = 7$, and $x = 3$.

Checking the Solutions

To check the solutions, we can plug each value of $x$ back into the original equation and verify that it is true.

For $x = 5$, we have $4(5)^3 - 72(5)^2 + 320(5) = 4(125) - 72(25) + 1600 = 500 - 1800 + 1600 = 300 \neq 420$.

For $x = 7$, we have $4(7)^3 - 72(7)^2 + 320(7) = 4(343) - 72(49) + 2240 = 1372 - 3528 + 2240 = -16 \neq 420$.

For $x = 3$, we have $4(3)^3 - 72(3)^2 + 320(3) = 4(27) - 72(9) + 960 = 108 - 648 + 960 = 420$.

Conclusion

Therefore, the only solution to the equation is $x = 3$. This means that the side length of the cut squares is $3$ inches.

The Dimensions of the Box

Now that we have found the side length of the cut squares, we can find the dimensions of the box.

The length and width of the box are $l - 2x$ and $w - 2x$, respectively. Since $x = 3$, we have $l - 2(3) = l - 6$ and $w - 2(3) = w - 6$.

The height of the box is $h$, which is equal to the original height of the sheet.

The Volume of the Box

The volume of the box is given by the formula $V = lwh$. Since we have found the dimensions of the box, we can plug in the values to find the volume.

We have $V = (l - 6)(w - 6)h$.

The Final Answer

Therefore, the final answer is $x = 3$, which represents the side length of the cut squares.

The dimensions of the box are $l - 6$, $w - 6$, and $h$, where $h$ is the original height of the sheet.

The volume of the box is $(l - 6)(w - 6)h$.

References

  • [1] "Algebraic Techniques for Solving Cubic Equations". MathWorld.
  • [2] "Cubic Equations". Wolfram MathWorld.
  • [3] "Volume of a Box". Math Open Reference.

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Introduction

In our previous article, we explored a mathematical problem involving a rectangular sheet of metal with identical squares cut from each corner. The sheet is then bent along the dotted lines to form an open box. We found that the side length of the cut squares is $3$ inches, and we determined the dimensions of the box.

In this article, we will answer some frequently asked questions about the problem and provide additional insights into the mathematical concepts involved.

Q&A

Q: What is the volume of the box?

A: The volume of the box is given by the formula $V = (l - 6)(w - 6)h$, where $l$, $w$, and $h$ are the length, width, and height of the box, respectively.

Q: How do we find the dimensions of the box?

A: To find the dimensions of the box, we need to know the original dimensions of the sheet. Let's assume that the original length and width of the sheet are $l$ and $w$, respectively. Then, the length and width of the box are $l - 2x$ and $w - 2x$, respectively, where $x$ is the side length of the cut squares.

Q: What is the relationship between the side length of the cut squares and the dimensions of the box?

A: The side length of the cut squares, $x$, is related to the dimensions of the box through the equations $l - 2x$ and $w - 2x$. As $x$ increases, the length and width of the box decrease.

Q: Can we find the original dimensions of the sheet?

A: Unfortunately, we cannot find the original dimensions of the sheet using the information given in the problem. However, we can find the dimensions of the box using the side length of the cut squares, $x$.

Q: What is the significance of the equation $4x^3 - 72x^2 + 320x = 420$?

A: The equation $4x^3 - 72x^2 + 320x = 420$ represents the volume of the box in terms of the side length of the cut squares, $x$. By solving this equation, we can find the value of $x$, which is the side length of the cut squares.

Q: How do we solve the equation $4x^3 - 72x^2 + 320x = 420$?

A: To solve the equation $4x^3 - 72x^2 + 320x = 420$, we can use various algebraic techniques, such as factoring and substitution. We can also use numerical methods, such as the Newton-Raphson method, to find the solution.

Q: What is the relationship between the side length of the cut squares and the volume of the box?

A: The side length of the cut squares, $x$, is related to the volume of the box through the equation $V = (l - 2x)(w - 2x)h$. As $x$ increases, the volume of the box decreases.

Additional Insights

  • The problem involves a rectangular sheet of metal with identical squares cut from each corner. This is a classic example of a geometric problem that can be solved using algebraic techniques.
  • The equation $4x^3 - 72x^2 + 320x = 420$ represents the volume of the box in terms of the side length of the cut squares, $x$. This equation can be solved using various algebraic techniques, such as factoring and substitution.
  • The side length of the cut squares, $x$, is related to the dimensions of the box through the equations $l - 2x$ and $w - 2x$. As $x$ increases, the length and width of the box decrease.
  • The volume of the box is given by the formula $V = (l - 2x)(w - 2x)h$, where $l$, $w$, and $h$ are the length, width, and height of the box, respectively.

Conclusion

In this article, we have answered some frequently asked questions about the problem and provided additional insights into the mathematical concepts involved. We have also discussed the relationship between the side length of the cut squares and the dimensions of the box, as well as the volume of the box.

We hope that this article has been helpful in understanding the mathematical concepts involved in the problem. If you have any further questions or need additional clarification, please do not hesitate to ask.