Pregunta
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  1. ( 2 points) A box with an open top is to be constructed from a square piece of cardboard with dimensions
    . This box is constructed by cutting out a square from each of the four corners and bending up
    the sides. Find the largest volume that such a box can have.

Ask by Bradley Powell. in the United States
Mar 14,2025

Solución de inteligencia artificial de Upstudy

Respuesta verificada por el tutor

Responder

The largest volume of the box is cubic centimeters.

Solución

Let be the side length of the square cut from each corner. After cutting, the dimensions of the base of the box will be
The height of the box is . Thus, the volume of the box is given by

Step 1. Express the Volume Function

Expanding the function:

Step 2. Find the Critical Points

Differentiate with respect to :
Set the derivative equal to zero:
Divide the equation by 2 to simplify:
Rearrange the terms:

Step 3. Solve the Quadratic Equation

Using the quadratic formula,
where , , and :
This gives two solutions:
Since leads to zero base dimensions (because ), it must be rejected. The feasible solution is:

Step 4. Calculate the Maximum Volume

Substitute into the volume function:
Simplify:
Thus,

Final Answer

The largest volume that the box can have is

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The Deep Dive

To maximize the volume of the box constructed from a square piece of cardboard, we can denote the side length of the squares cut from each corner as . After cutting out the corners and folding up the sides, the dimensions of the box will be:
  • Length =
  • Width =
  • Height =
The volume of the box can thus be expressed as:
To find the largest volume, we can take the derivative of the volume function with respect to :
  1. First, expand the volume function:
  1. Next, we find the derivative :
  1. Setting the derivative equal to zero to find critical points:
  1. Dividing through by 4 simplifies to:
  1. Using the quadratic formula:
This gives us the solutions:
Since would mean no box (height would be zero), we only consider .
  1. Finally, we plug back into the volume equation to find the maximum volume:
Thus, the largest volume that such a box can have is approximately .

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