2025 SAJC Promo Q11

A gardener wants to construct a greenhouse.

The greenhouse consists of two vertical glass walls \(ABHG\) and \(FEKL\) respectively, two slanted glass walls \(BCIH\) and \(EDJK\) respectively, and one glass ceiling \(CIJD\), as shown in Figure (1). Its cross-section \(ABCDEF\) is formed by rectangle \(ABEF\) and trapezium \(BCDE\), as shown in Figure (2).

Figure (1): Greenhouse

Figure (2): Cross-section of the greenhouse

The greenhouse has a length \(x\text{ m}\), width \(y\text{ m}\), and a rectangular floor area \(AGLF\) of \(100\text{ m}^2\).

The vertical glass walls are \(4\text{ m}\) tall, with the glass ceiling adding \(\frac{685}{x^2}\text{ m}\) to the height.

The width of the glass ceiling, \(CD\), is \(\frac{y}{2}\text{ m}\) wide, and the slant sides \(BC\) and \(DE\) are equal in length.

1. Show that \(S\), the total external surface area of the greenhouse that is made of glass is given by \(\left(8x+50+10\sqrt{\frac{18769}{x^2}+25}\right)\text{ m}^2\).[3]
2. Given that \(x=x_1\) is the value of \(x\) which gives a stationary value of \(S\), show that \(x_1\) satisfies the equation \(4x^2\sqrt{18769+25x^2}=93845\). Hence, find the value of \(x_1\) to 3 decimal places, and show that \(S\) is a minimum at this value.[6]
3. Hence, given that the cost to install the glass is \(\$200\) per \(\text{m}^2\), estimate the minimum cost of installing the glass to the nearest dollar.[2]
4. Sketch the graph of \(S\) as \(x\) varies, labelling the coordinates of the minimum point and equations of asymptotes.[2]