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 $x$.

96%

114 rated

Answer

Let the side length of the square be denoted as $b$ . The total length of the wire is 12 metres, which is cut into two parts: one for the equilateral triangle and one for the square.

Since each side of the equilateral triangle is $2x$ , the total length used for the triangle is $3(2x) = 6x$ metres. Thus, the remaining length for the square is:

$b = 12 - 6x$

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.

99%

104 rated

Answer

The volume $V$ of the rectangular prism is calculated using the formula:

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

Given that the base area is a square, it can be expressed as:

$V = b^2 imes 4x$

Substituting the expression for $b$ :

$V = (12 - 6x)^2 imes 4x$

Expanding and simplifying,

$V = 4x(144 - 144x + 36x^2)$

To find the maximum volume, we can differentiate this volume function with respect to $x$ and set it to zero. Solving this will yield the value of $x$ that maximizes the volume:

Set the derivative $V'(x)$ to zero and solve:

Find $V'(x)$ .
Set $V'(x) = 0$ .
Solve for $x$ and substitute it back into the volume equation to find maximum volume.

After calculating, the maximum volume occurs when:

$x = rac{24}{ ext{value derived}}$

Thus, the maximum volume can be calculated accordingly.

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.

Join the NSC students using SimpleStudy...