A box open from top is made from a rectangular sheet of dimension a × b by cutting squares each of s

Q: Wait, why is the length $(a-2x)$? Shouldn't it just be $(a-x)$?

This is the most common trap! Imagine the sheet. You are cutting a square of side $x$ from the *left* corner AND the *right* corner. So, you are removing $x$ twice from the total length $a$. That leaves you with $a - x - x = a - 2x$. Always draw the diagram to visualize the two cuts!

Q: Why do we reject the root with the plus sign in the quadratic formula?

Great observation! Mathematically, both roots satisfy the derivative equation. However, we are building a physical box. If $x$ is too large, the flaps would overlap or consume the entire sheet, leaving no base. Since $x$ must be less than half of the shorter side (i.e., $x < \frac{b}{2}$), the larger root violates this physical reality. We only keep the root that fits within the sheet's dimensions.

Q: How do I know this is a maximum and not a minimum?

Excellent mathematical rigor! To be absolutely sure, you should perform the second derivative test. Find $V''(x)$. If $V''(x) < 0$ at your critical point, it is a local maximum. For this cubic function, the smaller root will yield a negative second derivative, confirming it is indeed the maximum volume.

Question

A box open from top is made from a rectangular sheet of dimension $a 	imes b$ by cutting squares each of side $x$ from each of the four corners and folding up the flaps. If the volume of the box is maximum, then $x$ is equal to :

Options:

(A) $\frac{a+b-\sqrt{a^2+b^2-ab}}{12}$

(B) $\frac{a+b-\sqrt{a^2+b^2+ab}}{6}$

(C) $\frac{a+b-\sqrt{a^2+b^2-ab}}{6}$

(D) $\frac{a+b+\sqrt{a^2+b^2-ab}}{6}$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing at your desk with a rectangular sheet of metal. Its dimensions are $a$ and $b$. You have a pair of scissors, and your goal is to create an open-top box.

To do this, you must cut out identical squares of side $x$ from each of the four corners. These cutouts allow you to fold up the remaining flaps, making the height of your box exactly $x$.

The original length $a$ is shortened by $x$ on the left and $x$ on the right, so the new length is $L = a - 2x$. Similarly, the width $b$ is shortened by $x$ on the top and $x$ on the bottom, resulting in a new width $W = b - 2x$.

We have successfully mapped our physical reality into algebraic variables: $L = a - 2x$, $W = b - 2x$, and $H = x$.

### The Volume Function

We want to maximize the volume $V$. The volume of a cuboid is the product of its length, width, and height.

$$V(x) = (a - 2x)(b - 2x)x$$

To make it easier to handle, let us expand this expression. Multiplying the terms, we get $V(x) = (ab - 2ax - 2bx + 4x^2)x$, which simplifies to the cubic polynomial:

$$V(x) = 4x^3 - 2(a + b)x^2 + abx$$

This polynomial represents the volume for any given cut size $x$. Our goal is to find the specific $x$ that pushes this volume to its absolute peak.

### The Calculus of Change

In the language of calculus, a maximum occurs where the rate of change of the function is zero. We need to find the derivative of $V(x)$ with respect to $x$ and set it to zero.

Applying the power rule, we differentiate:

$$V'(x) = \frac{d}{dx}(4x^3 - 2(a + b)x^2 + abx) = 12x^2 - 4(a + b)x + ab$$

Setting $V'(x) = 0$, we arrive at the quadratic equation:

$$12x^2 - 4(a + b)x + ab = 0$$

This is the heart of the problem. We have a quadratic equation in the form $Ax^2 + Bx + C = 0$, where $A = 12$, $B = -4(a+b)$, and $C = ab$.

### Solving the Quadratic

We invoke the quadratic formula: $x = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A}$. Substituting our coefficients, we get:

$$x = \frac{4(a+b) \pm \sqrt{[-4(a+b)]^2 - 4(12)(ab)}}{24}$$

Let us simplify the discriminant $\Delta$:

$$\Delta = 16(a+b)^2 - 48ab = 16(a^2 + 2ab + b^2) - 48ab = 16(a^2 - ab + b^2)$$

Taking the square root of the discriminant, we get $4\sqrt{a^2 - ab + b^2}$. Substituting this back into our formula for $x$:

$$x = \frac{4(a+b) \pm 4\sqrt{a^2 - ab + b^2}}{24}$$

Dividing the numerator and denominator by 4, we find the two critical points:

$$x = \frac{(a+b) \pm \sqrt{a^2 + b^2 - ab}}{6}$$

### The Physical Reality

We have two mathematical solutions, but only one makes sense in the physical world. Remember, $x$ is the side of the square we cut out.

If we choose the root with the plus sign, $x$ becomes too large, exceeding the physical dimensions of the sheet. Therefore, we must reject the positive root.

The only valid solution is the one with the negative sign:

**$$x = \frac{a+b - \sqrt{a^2 + b^2 - ab}}{6}$$**

This is the optimal cut size. You have successfully navigated the geometry, the algebra, and the calculus to find the perfect dimension.

Animated Solution for Mathematics - Differentiation: A box open from top is made from a rectangular sheet of dimension $a \times b$ by cutting squares each of side $x$ from each of the four corners and folding up the flaps. If the volume of the box is maximum, then $x$ is equal to :

Select Answer:

Visualized Solution

Visualizing the Sheet

Cutting the Corners

Folding the Flaps

Dimensions of the Box

The Volume Function

Expanding the Expression

Condition for Maxima

Differentiating $V (x)$

Applying the Quadratic Formula

Simplifying the Discriminant

Expanding the Discriminant

Finding the Roots

Simplified Roots

Selecting the Valid Root

Final Conclusion

The Sigma Insight: Maxima and Minima

Analyzing the Setup

The Volume Function

The Calculus of Change

Solving the Quadratic

The Physical Reality

Similar Questions

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Determine the points of maxima and minima of the function $f (x) = \frac{1}{8} ln x - b x + x^{2}, x > 0$ , where $b \geq 0$ is a constant.

Select Answer:

Visualized Solution

Visualizing the Sheet

Cutting the Corners

Folding the Flaps

Dimensions of the Box

The Volume Function

Expanding the Expression

Condition for Maxima

Differentiating .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } V(x)

Applying the Quadratic Formula

Simplifying the Discriminant

Expanding the Discriminant

Finding the Roots

Simplified Roots

Selecting the Valid Root

Final Conclusion

The Sigma Insight: Maxima and Minima

Analyzing the Setup

The Volume Function

The Calculus of Change

Solving the Quadratic

The Physical Reality

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