A square sheet of cardboard, of side 10 units, is used to make an open box. Squares of side x units, where x ∈ ℝ, are removed from each corner of the cardboard and it is then folded along the dotted lines, as shown in the diagram below, in order to create the box. (a) The length (l), breadth (b) and height (h) of the box are shown in the diagram above. Write l, b, and h in terms of x. l = b = h = (b) Show that the volume of the box can be written as V(x) = (c) Explain why a box of height 6 units cannot be made from the sheet of cardboard.

Leaving Cert Mathematics Algebra: A square sheet of cardboard, of

A square sheet of cardboard, of side 10 units, is used to make an open box - Leaving Cert Mathematics - Question 8 - 2021

Question 8

A square sheet of cardboard, of side 10 units, is used to make an open box. Squares of side x units, where x ∈ ℝ, are removed from each corner of the cardboard and ... show full transcript

Worked Solution & Example Answer:A square sheet of cardboard, of side 10 units, is used to make an open box - Leaving Cert Mathematics - Question 8 - 2021

Step 1

a) The length (l), breadth (b) and height (h) of the box are shown in the diagram above. Write l, b, and h in terms of x.

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Answer

The dimensions of the box can be defined in terms of the variable x:

The length (l) of the box will be equal to the original length of the cardboard minus twice the side of the squares cut out:
$l = 10 - 2x$
The breadth (b) follows the same logic:
$b = 10 - 2x$
The height (h) of the box is simply the side of the squares cut out, which is:
$h = x$

Step 2

b) Show that the volume of the box can be written as V(x) = 4x^3 - 40x^2 + 100x.

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Answer

To find the volume (V) of the box, we can use the formula:
$V = l imes b imes h$
Substituting the expressions for l, b, and h from part (a):
$V = (10 - 2x)(10 - 2x)(x)$
Expanding this, we first multiply the two expressions:
$= (10 - 2x)(10 - 2x) = 100 - 40x + 4x^2$
Now, multiplying by x:
$V = x(100 - 40x + 4x^2)$
$= 100x - 40x^2 + 4x^3$
By rearranging, we can express this as:
$V(x) = 4x^3 - 40x^2 + 100x$ .

Step 3

c) Explain why a box of height 6 units cannot be made from the sheet of cardboard.

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Answer

To have a box of height 6 units, we need x = 6. Using the formula for the dimensions, we find the width:
$b = 10 - 2x = 10 - 2(6) = 10 - 12 = -2$
Since a negative width does not make sense in this context, it is concluded that a height of 6 units cannot be obtained, as it results in a negative dimension for the box.

A square sheet of cardboard, of side 10 units, is used to make an open box - Leaving Cert Mathematics - Question 8 - 2021

Worked Solution & Example Answer:A square sheet of cardboard, of side 10 units, is used to make an open box - Leaving Cert Mathematics - Question 8 - 2021

a) The length (l), breadth (b) and height (h) of the box are shown in the diagram above. Write l, b, and h in terms of x.

b) Show that the volume of the box can be written as V(x) = 4x^3 - 40x^2 + 100x.

c) Explain why a box of height 6 units cannot be made from the sheet of cardboard.

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