Answered

Is it possible to cut the squares so that the volume of the box is $40\text{ in.}^3$?

Find all real solutions of this equation to answer the question.

$(6-2x)(3-2x)x=40$


AYes. Because $\frac{3}{2}$ is a root, you can cut squares with sides of $\frac{3}{2}\text{ in.}$ to make the box.
BNo. The only real solution is $x=4$. It is not possible to cut squares of this size.
CNo. This equation has no real solutions.

Verified Answer :

  • The volume equation \((6-2x)(3-2x)x = 40\) leads to the cubic equation \(2x^3 - 9x^2 + 9x - 20 = 0\).
  • Physical constraints require \(0 < x < 1.5\).
  • The only real root of the equation is \(x=4\), which falls outside the required interval.
  • It is therefore impossible to construct such a box, confirming that option (b) is the correct answer.
  • \(\boxed{\text{b}}\)

Explanation

  1. Analysis of the Problem and Constraints

To determine if it is possible to cut squares of side length \(x\) from the corners of a rectangular sheet to create a box with a volume of \(40 \text{ in}^3\), we analyze the given volume equation:
\[(6-2x)(3-2x)x = 40\]
For the dimensions of the resulting box to be physically valid, all side lengths must be positive. This gives us the constraints:

  1. \(x > 0\)
  2. \(6 - 2x > 0 \implies x < 3\)
  3. \(3 - 2x > 0 \implies x < 1.5\)

Combining these constraints, we must find a real solution \(x\) such that \(0 < x < 1.5\).

  1. Solving the Cubic Equation

Next, we expand the volume equation to solve for \(x\):
\[(6 - 2x)(3 - 2x)x = 40\]
First, multiply the binomials:
\[(18 - 12x - 6x + 4x^2)x = 40\]
\[(4x^2 - 18x + 18)x = 40\]
\[4x^3 - 18x^2 + 18x = 40\]
\[4x^3 - 18x^2 + 18x - 40 = 0\]
Dividing the entire equation by 2 to simplify:
\[2x^3 - 9x^2 + 9x - 20 = 0\]
We solve this cubic equation to find the roots. By testing potential rational roots or solving the equation directly, we find that \(x=4\) is a root:
\[2(4)^3 - 9(4)^2 + 9(4) - 20 = 2(64) - 9(16) + 36 - 20 = 128 - 144 + 36 - 20 = 0\]
Factoring out \((x-4)\) from \(2x^3 - 9x^2 + 9x - 20\):
\[(x-4)(2x^2 - x + 5) = 0\]
The quadratic factor \(2x^2 - x + 5 = 0\) has no real roots, as the discriminant \(\Delta = (-1)^2 - 4(2)(5) = 1 - 40 = -39 < 0\). Thus, the only real solution is \(x=4\).

  1. Conclusion and Final Answer

We found the only real solution to be \(x=4\). Checking this against our constraints from Step 1, we see that the physical construction of the box requires \(0 < x < 1.5\). Since \(x=4\) does not fall within the interval \((0, 1.5)\), it is impossible to cut squares of side length \(x\) to achieve a volume of \(40 \text{ in}^3\). Therefore, option (b) is the correct conclusion.

Examples

In engineering and manufacturing, this type of optimization problem is common when designing packaging or containers. By modeling the volume of a box as a function of the material removed, engineers can determine if a desired capacity is physically feasible given the dimensions of the initial raw material sheet. If the math shows that the required dimensions cannot be achieved, the design must be revised.