A wire, 12 metres long, is cut into two pieces. One part is bent to form an equilateral triangle and the other a square. A side of the triangle has a length of $2x$ metres. 9.1 Write down the length of a side of the square in terms of $x$. 9.2 If this square is now used as the base of a rectangular prism with a height of $4x$ metres, determine the maximum volume of the rectangular prism.

NSC Mathematics Euclidean Geometry: A wire, 12 metres long, is cut i

A wire, 12 metres long, is cut into two pieces - NSC Mathematics - Question 9 - 2023 - Paper 1

Question 9

A wire, 12 metres long, is cut into two pieces. One part is bent to form an equilateral triangle and the other a square. A side of the triangle has a length of $2x$ ... show full transcript

Worked Solution & Example Answer:A wire, 12 metres long, is cut into two pieces - NSC Mathematics - Question 9 - 2023 - Paper 1

Step 1

Write down the length of a side of the square in terms of $x$.

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Answer

The total wire length is 12 metres. The side length of the equilateral triangle is $2x$ metres, so the perimeter of the triangle is $3(2x) = 6x$ metres. The remaining wire is used to form a square, whose perimeter can be calculated as:

Perimeter of the square = Total length - Perimeter of triangle

$ext{Perimeter of the square} = 12 - 6x$

Since the perimeter of a square is $4$ times the side length, let the length of a side of the square be denoted as $s$ :

$4s = 12 - 6x$

Solving for $s$ , we get:

$s = rac{12 - 6x}{4} = 3 - rac{3}{2}x$ .

Thus, the length of a side of the square in terms of $x$ is:

$s = 3 - rac{3}{2}x$ .

Step 2

If this square is now used as the base of a rectangular prism with a height of $4x$ metres, determine the maximum volume of the rectangular prism.

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Answer

The volume $V$ of a rectangular prism is given by the formula:

$V = ext{Base Area} imes ext{Height}$

For our case, the base area (area of the square) is $s^2$ and the height is $4x$ . Thus,

$V = s^2(4x)$
Substituting the expression for $s$ :

$V = igg(3 - rac{3}{2}xigg)^2(4x)$

Now expanding this expression:

$V = (9 - 9x + rac{9}{4}x^2)(4x) = 36x - 36x^2 + 9x^3$

To find the maximum volume, we take the derivative of $V$ and set it to zero:

$V'(x) = 36 - 72x + 27x^2$

Setting the derivative equal to zero:

$27x^2 - 72x + 36 = 0$
Dividing by 9 gives:

$3x^2 - 8x + 4 = 0$
Using the quadratic formula $x = rac{-b ext{±} ext{√}(b^2 - 4ac)}{2a}$ , where $a = 3$ , $b = -8$ , and $c = 4$ :

$x = rac{8 ext{±} ext{√}((-8)^2 - 4 imes 3 imes 4)}{2 imes 3} = rac{8 ext{±} ext{√}(64 - 48)}{6} = rac{8 ext{±} ext{√}16}{6}$

Which gives:

$x = rac{8 ext{\pm} 4}{6}$
Calculating the two possible values for $x$ :

$x = rac{12}{6} = 2$
$x = rac{4}{6} = rac{2}{3}$

Substituting back into our volume formula to find the maximum volume at $x = 2$ :

$V = 36 imes 2 - 36 imes (2)^2 + 9 imes (2)^3 = 72 - 144 + 72 = 0$

For $x = rac{2}{3}$ :

$V = 36 imes rac{2}{3} - 36 imes igg( rac{2}{3}igg)^2 + 9 imes igg( rac{2}{3}igg)^3 = 24 - 16 + 8 = 16$

Thus, the maximum volume of the rectangular prism is approximately:

$V = rac{32}{3} ext{ m}^3 ext{ or } 10.67 ext{ m}^3$ .

A wire, 12 metres long, is cut into two pieces - NSC Mathematics - Question 9 - 2023 - Paper 1

Worked Solution & Example Answer:A wire, 12 metres long, is cut into two pieces - NSC Mathematics - Question 9 - 2023 - Paper 1

Write down the length of a side of the square in terms of $x$.

If this square is now used as the base of a rectangular prism with a height of $4x$ metres, determine the maximum volume of the rectangular prism.

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