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.

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justaguide | College Teacher | (Level 2) Distinguished Educator

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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.

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