Question

A box with no top is to be constructed from a piece of cardboard whose length measures 12 inches more than its width. The box is formed by cutting squares that measure 4 inch on each side from the four corners and then folding up the sides. If the volume of the box 256 in cm^3 what are the dimensions of the piece of cardboard? The Height? The width?

Answer 1

Answer:

Width is 12 inches and height is 4 inches.

Step-by-step explanation:

Let the width of the piece of cardboard be = w inches

Then length is = $w+12$ inches

As given, 4 inch squares are cut from each corner;

So, the width is = $w-8$ inches

Length is = $w+12-8=w+4$ inches

When we are folding the cut portion, that gives the height as = 4 inches

The volume is given by : length x width x height

And volume is given as 256 cubic inches

So, $256=(w+4)(w-8)(4)$

=> $256=(w^{2}-4w-32)4$

=> $64=(w^{2}-4w-32)$

=> $w^{2} -4w=96$

or $w^{2} -4w-96=0$

=> $w^{2} -12w+8w-96=0$

=> $w(w-12)+8(w-12)=0$

We get , $(w+8)=0$ ; w = -8(neglect the negative value)

$(w-12)=0$ ; w = 12 inches

And length = $12+12=24$ inches

Hence, dimensions are : L = 24 inches ; w = 12 inches and h = 4 inches

We can check by putting these values in equation : $256=(w+4)(w-8)(4)$

$256=(12+4)(12-8)(4)$

=> $256=(16)(4)(4)$

=> $256=256$

Answer 2

The dimensions of the carboard are related to the volume of the box

formed and the required cut out squares.

Dimensions of the cardboard are;

• The cardboard has a width of 12 inches.

• Height of the cardboard is 24 inches.

Reasons:    

The given parameters are;

Length of the cardboard = The width of the cardboard + 12 inches

Side length of cut out squares = 4 inches

Volume of the box = 256 in³ (corrected to compatible units and based on a similar question)

Required:

The dimension (height and width) of the piece of cardboard

Solution;

Let W represent the width of the cardboard, and H the height, we have;

H = W + 12

The width of the box = W - 2×4 = W - 8

The length of the box = H - 2×4 = W + 12 - 2 × 4 = W + 4

The height of the box = 4 inches

Volume of the box, V = Width of box × Length of box × Height of box

Therefore;

V = (W - 8)·(W + 4) × 4 = 4·W² - 16·W - 128

V = 256 in.³

∴ V = 4·W² - 16·W - 128 = 256

4·W² - 16·W - 128 - 256 = 0

Which gives;

(W - 12)·(W + 8)·4 = 0

Which gives;

W = 12

• The width of the cardboard, W = 12 inches

H = W + 12

∴ H = 12 + 12 = 24

• The height of the cardboard, H = 24 inches

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