What is the maximum number of plants that can be grown in a farm of 100 m^2 if a plant needs a minimum 3 m^2 and with each additional plant, the space per plant decreases by 0.1 m^2.
The space required by a plant decreases by 0.1 m^2 per plant as the number of plants is increased. The minimum area required by a plant is 3 m^2. The maximum number of plants that can be grown in a farm of 100 m^2 has to be determined.
Let us start with one plant in the farm. The area available for the plant is 100^2. If the number doubles, the area required per plant decreases by 0.1 m^2 or each plant requires (50 - 0.1) m^2.
If n plants are planted in the area, the required area is (100/n - (n -1)*0.1)*n
This cannot be less than 3
=> (100/n - (n - 1)*0.1) > 3
=> `100/n - n*0.1 + 0.1 > 3`
=> `100 - 0.1n^2 - 2.9n > 0`
Solve 100 - 0.1*n^2 - 2.9*n = 0
The root less than 0 is approximately 49.28
The maximum number of plants that can be grown in the farm is 49.