Question

A open top box is to be created from a piece of cardboard measuring 26 inches by 35 inches by cutting out identical squares with side length of x from each corner and folding up the sides. Express the surface area A of the box as a function of x?

Answer 1

The surface area A of the open-top box can be expressed as a function of x by using the equation A(x) = -7x^2 + 140x.

Explanation:

To express the surface area A of the open-top box as a function of x, we need to calculate the area of each face and sum them up. The open-top box is formed by cutting out identical squares with side length x from each corner of a piece of cardboard measuring 26 inches by 35 inches.

The dimensions of the box can be visualized as follows:

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| x |

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| x | | x |

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| x |

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The surface area A can be expressed as a function of x using the following formula:

A(x) = x^2 + 4xh

To find the value of h, we need to subtract twice the value of x from the original dimensions of the cardboard:

h = 35 - 2x

Substituting this value into the formula, we get:

A(x) = x^2 + 4x(35 - 2x)

Simplifying further, we have:

A(x) = x^2 + 140x - 8x^2

Combining like terms, we get:

A(x) = -7x^2 + 140x

Therefore, the surface area A of the open-top box can be expressed as a function of x by using the equation A(x) = -7x^2 + 140x.

Answer 2

To find the surface area A of the box, we need to determine the area of the six faces that make up the box.

1. Top and bottom faces:

- The top and bottom faces of the box will be rectangles with dimensions (35-2x) inches by (26-2x) inches.

- The area of one top or bottom face is given by length × width.

- So, the total area of the top and bottom faces is 2 × (35-2x) × (26-2x) square inches.

2. Side faces:

- The side faces of the box will be rectangles with dimensions (35-2x) inches by x inches or (26-2x) inches by x inches, depending on the orientation of the box.

- The area of one side face is given by length × width.

- So, the total area of the side faces is 2 × (35-2x) × x + 2 × (26-2x) × x square inches.

3. Total surface area:

- The total surface area of the box is the sum of the areas of the top, bottom, and side faces.

- So, the total surface area A can be expressed as:

A = 2 × (35-2x) × (26-2x) + 2 × (35-2x) × x + 2 × (26-2x) × x

Therefore, the surface area A of the box, as a function of x, is given by the equation A = 2 × (35-2x) × (26-2x) + 2 × (35-2x) × x + 2 × (26-2x) × x.

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