Alex is starting a vertical farming business in Singapore, using hydroponics to grow leafy vegetables in an indoor warehouse space of floor area \(50\text{ m}^2\) with a ceiling height of 2.5 metres. The current setup consists of 8 vertical hydroponic shelves arranged within the space.
Fixed overheads and costs: electricity is $0.25 per kWh, water is $2.50 per \(\text{m}^3\), nutrients cost $0.30 per kg of vegetables grown, and the business rents a government-subsidised commercial unit at a special rate of $500 per month.
The table below gives information on the monthly production, unit price and resource usage for January to June 2025.
Resource Usage| Month | Monthly production (kg) | Price (\(\$/\text{kg}\)) | Electricity (kWh) | Water (m³) |
|---|
| Jan | \(240\) | \(6.00\) | \(800\) | \(15\) |
| Feb | \(260\) | \(6.10\) | \(820\) | \(16\) |
| Mar | \(280\) | \(6.20\) | \(840\) | \(17\) |
| Apr | \(300\) | \(6.30\) | \(860\) | \(18\) |
| May | \(320\) | \(6.40\) | \(880\) | \(19\) |
| Jun | \(340\) | \(6.50\) | \(900\) | \(20\) |
Calculate the profit for the month of June.[3]
Suggest a possible reason for the increase in crop prices from January to June.[1]
The vertical farming system uses hydroponic shelves, each measuring \(2.0\text{ m}\) (length) \(\times0.5\text{ m}\) (width) \(\times1.8\text{ m}\) (height), with 5 levels of planting trays per shelf. Each level grows 10 kg of vegetables per month. There are currently 8 such shelves arranged in the warehouse. Safety guidelines require a minimum clearance of 0.2 m from the ground and 0.5 m from the ceiling. The maximum number of shelves that can be installed is limited by both walking and maintenance space requirements. Each shelf occupies \(2.0\text{ m}\times0.5\text{ m}\) of floor space, and minimum clear walking space of 0.5 m must be provided around each shelf. This clearance applies to the warehouse walls and between shelves. The warehouse has a total floor area of \(50\text{ m}^2\) (\(10\text{ m}\times5\text{ m}\)). Due to airflow and light distribution limits, the maximum number of shelves that can physically fit in the warehouse is limited. For 10 shelves or fewer, each shelf operates at 100% of its production capacity. When 11 or more shelves are installed, airflow saturation occurs, reducing the production of all shelves to 85% of their monthly production. In addition, each shelf beyond the 10th operates at only 70% of its monthly production due to poor light and air mixing at the edges. Alex’s goal is to maximise revenue for the month of July. Assume that the unit price and resource usage for July is the same as June. Determine the number of shelves Alex should install. Justify your answer clearly with mathematical calculations that consider both space constraints and the productivity limitations. You may wish to draw diagrams to aid in your understanding.[6]