10 A piece of wire of length 66 cm is bent to form the five sides of a pentagon.
The pentagon consists of three sides of a rectangle and two sides of an equilateral triangle.
The sides of the rectangle measure \(x \mathrm {~cm}\) and \(y \mathrm {~cm}\) and the sides of the triangle measure \(x \mathrm {~cm}\), as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{e3635007-2ad1-4b2a-b937-41fe90bb1111-12_405_492_630_863}
10
- You are given that \(\sin 60 ^ { \circ } = \frac { \sqrt { 3 } } { 2 }\)
Explain why the area of the triangle is \(\frac { \sqrt { 3 } } { 4 } x ^ { 2 }\)
10
- (ii) Show that the area enclosed by the wire, \(A \mathrm {~cm} ^ { 2 }\), can be expressed by the formula
$$A = 33 x - \frac { 1 } { 4 } ( 6 - \sqrt { 3 } ) x ^ { 2 }$$
10
- Use calculus to find the value of \(x\) for which the wire encloses the maximum area. Give your answer in the form \(p + q \sqrt { 3 }\), where \(p\) and \(q\) are integers.
Fully justify your answer.
\(L _ { 1 }\) is a tangent to the circle \(C\) at the point \(P ( 6,5 )\)
The line \(L _ { 2 }\) has equation \(y = x + 3\)
\(L _ { 2 }\) is a tangent to the circle \(C\) at the point \(Q ( 0,3 )\)
The lines \(L _ { 1 }\) and \(L _ { 2 }\) and the circle \(C\) are shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{e3635007-2ad1-4b2a-b937-41fe90bb1111-14_702_714_726_753}