Lina's company sells a type of stained glass as shown in the diagram above. It is in the shape of a

Question

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. 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. 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 $\$P$. Show that $P = 120 - \frac{3}{5}t - \frac{280}{t}$.[1] 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. Find the possible set of values of $t$ for which Lina makes a profit of at least $\$80$ per piece of stained glass.

Accepted Answer

(b)max value of A =$\frac{18}{\pi +8-2\sqrt 3}$ (e)$8 \leq t \leq 58$

2024 SAJC H1 Q5

2024 SAJC H1 Q5