2024 SAJC H1 Q5

Lina's company sells a type of stained glass as shown in the diagram above. It is in the shape of a rectangle \(ABCD\) with a semicircle joined to \(AB\) as its diameter and an equilateral triangle joined at \(DC\). The length of \(AB\) is \(2x\) m and the length of \(BC\) is \(y\) m. The total perimeter of the single piece of glass is \(6\) m.

1. Show that the area, \(A\) m, of each piece of stained glass is given by\( A = 6x - \frac{1}{2}x^2 ( \pi + 8 - 2\sqrt{3} ) m^2\).[4]

A customer wants to buy a piece of stained glass with the largest possible area.

2. Use a non-calculator method to find the maximum exact value of \(A\). Justify that this is a maximum value.[4]

Lina develops a model for the cost of producing the stained glasses. In her model, the cost of producing \(t\) pieces of stained glass per day is \(\frac{3}{5}t^2 + 130t + 280\). Lina sells stained glass in packets of \(4\) pieces for \(\$1000\) per packet. Using this model, the profit Lina makes for each piece of stained glass is denoted by \(\$P\).

3. Show that \(P = 120 - \frac{3}{5}t - \frac{280}{t}\).[1]
4. Sketch the graph of \(P\) against \(t\), stating the coordinates of any intersection point(s) with the horizontal axis, equation of the asymptote(s) and the coordinates of the turning point.
5. Find the possible set of values of \(t\) for which Lina makes a profit of at least \(\$80\) per piece of stained glass.