A candy box is to be made out of a piece of cardboard that measures 8 inches by 12 inches. Squares of equal size have to be cut out of each corner, so that the sides are folded up... A candy box is to be made out of a piece of cardboard that measures 8 inches by 12 inches. Squares of equal size have to be cut out of each corner, so that the sides are folded up to form a rectangular box with open top. What square size should be cut from each corner to obtain a maximum volume?
Understand the Problem
The question is asking for the size of square corners that should be cut from a piece of cardboard to maximize the volume of an open-top rectangular box formed by folding up the sides. It involves determining the optimal 'x' value where squares of size 'x' are removed from each corner.
Answer
The optimal size to cut is approximately $x \approx 1.57$ inches.
Answer for screen readers
The optimal square size that should be cut from each corner is approximately: $$ x \approx \frac{10 - 2\sqrt{7}}{3} \approx 1.57 \text{ inches} $$
Steps to Solve
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Define the dimensions of the box
The piece of cardboard measures 8 inches by 12 inches. After cutting out squares of side length $x$ from each corner, the new dimensions of the base of the box will be:
- Length = $12 - 2x$ (subtracting $x$ from both ends)
- Width = $8 - 2x$
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Write the volume formula
The volume $V$ of the open-top box can be expressed as: $$ V = \text{Length} \times \text{Width} \times \text{Height} $$ Where the height is $x$. Thus, the volume becomes: $$ V = (12 - 2x)(8 - 2x)(x) $$
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Expand the volume expression
Expand the equation: $$ V = x(12 - 2x)(8 - 2x) $$ First, expand $(12 - 2x)(8 - 2x)$: $$ V = x((96 - 24x - 16x + 4x^2)) $$ This simplifies to: $$ V = x(96 - 40x + 4x^2) $$ Further expanding gives: $$ V = 96x - 40x^2 + 4x^3 $$
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Find the derivative and critical points
Calculate the first derivative of the volume: $$ \frac{dV}{dx} = 96 - 80x + 12x^2 $$ Set the derivative to zero to find critical points: $$ 12x^2 - 80x + 96 = 0 $$
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Solve the quadratic equation
Use the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a = 12$, $b = -80$, and $c = 96$:
- Calculate the discriminant: $$ b^2 - 4ac = (-80)^2 - 4(12)(96) = 6400 - 4608 = 1792 $$
- Solve for $x$: $$ x = \frac{80 \pm \sqrt{1792}}{24} $$
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Determine maximum volume
Evaluate the two solutions from the quadratic formula: $$ x = \frac{80 + \sqrt{1792}}{24} \quad \text{and} \quad x = \frac{80 - \sqrt{1792}}{24} $$ Determine which value gives a maximum by testing values or employing the second derivative test.
The optimal square size that should be cut from each corner is approximately: $$ x \approx \frac{10 - 2\sqrt{7}}{3} \approx 1.57 \text{ inches} $$
More Information
Cutting squares of approximately 1.57 inches from each corner allows the box to achieve maximum volume. This scenario is a classic optimization problem in calculus that utilizes derivatives to find maximum values.
Tips
- Neglecting Constraints: Ensure that $x$ is within reasonable bounds (i.e., $x$ must be less than 4 inches due to the box dimensions).
- Incorrect Expansion: When expanding the volume formula, be careful with each term to avoid arithmetic mistakes.
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