Take the above problem. I understand how to do the calculation. However I can't seem to understand the units of the answer in the calculation.
The answers illustrate the units as t/ha^-2 which is equivalent to t(ha)^2. But where do these units arise from? Crop yield is represented by tones per ha in the table above. So wouldn't total crop yield just be tones? I can't seem to find any intuition whatsoever as to why the units are t/ha^-2? Could someone please give me a hint or a solution?
I think this is a mistake in the provided solutions – it should say either $\mathrm{t/ha}$ or $\mathrm{t \, ha}^{-1}$. Furthermore, the answer should be multiplied with the area of fields.
There are two different quantities in this task: yield which is the mass of harvested crops (and has units $\mathrm{t}$), and field yield which is yield per unit of field area (which has units $\mathrm{t/ha}$). In the provided table, you are given field yield, but in the question, the farmer wants to have the highest yield.
The yield if fertilizer is put on one field will be: $$m=7\;{\rm \frac{t}{ha}}\times 1\;{\rm ha}+3\;{\rm \frac{t}{ha}}\times 1\;{\rm ha}=10\;{\rm t}.$$
The yield if the fertilizer is spread equally on both fields will be: $$m=4.8\;{\rm \frac{t}{ha}}\times 1\;{\rm ha}+4.8\;{\rm \frac{t}{ha}}\times 1\;{\rm ha}=9.6\;{\rm t}.$$
Therefore, the farmer will produce more crops if he puts all fertilizer on one field.
