Hi, I just need help on the last question!

A farmer is growing winter wheat. The amount of wheat he will get per hectare depends on, among other things, the amount of nitrogen fertiliser that he uses. For his particular farm, the amount of wheat depends on the nitrogen in the following way: Y = 7000+ 32N - 0.1N², where Y is the amount of wheat produced, in kg per hectare, and N is the amount of nitrogen added, in kg per hectare.

(i) How much wheat will he get per hectare if he uses 100 kg of nitrogen per hectare?
(ii) Find the amount of nitrogen that he must use in order to maximise the amount of wheat produced.
(iii) What is the maximum possible amount of wheat produced per hectare?
(iv) The farmer's total costs for producing the wheat are €1300 per hectare. He can sell the wheat for €160 per tonne. He can also get €75 per hectare for the leftover straw. If he achieves the maximum amount of wheat, what is his profit per hectare?

To find the maximum, we find first derivative. Doing this is quite of easy,just subtract the exponent by 1(Just when there will be a term in the combination, suppose 5*x)and multiply the number by the original exponent.

Here

d/dx = (0*7000*x⁰) + 1(32)*N¹-¹ - 2*0.1 N²-¹
d/dx = 0 + 32 - 0.2N = 32-0.2N

Take second derivative by taking into account first derivative(Derivative of 32-0.2N)

d²/dx² = 0*32*x⁰ - 1(0.2*N¹-¹) = 0-0.2 = - 0.2 (-ve,not possible)

We now equate the first derivative with 0:

d/dx = 0
32-0.2N = 0
32 = 0.2N
N = 32*10/2 = 160

We take into consideration N = 160 since of we see that the second derivative's result is negative.

In order to maximize the amount of wheat produced, we see the nitrogen he must use is 160 kg/hectare.

(iii)What is the maximum possible amount of wheat produced per hectare?

Maximum possible amount of wheat is when N = 160 kg/hectare.

Thus put N = 160

Y = 7000+160*32 + 160²*(-0.1) =
Y = 7000+ 5120 -2560
Y = 12120 -2560
Y = 9560 kg/hectare

(iv)Note that he achieved maximum amount of wheat,i.e. 9560 kg/hec

Given that total costs for producing wheat: €1300/hectares

He sold the wheat for €160 / tonnes

Note:

1 tonne = 1000kg

So maximum amount of wheat = 9560/1000 tonnes = 9.56 tonnes

1 tonne ---> €160

Thus as he sold the maximum amount of wheat = 9.56 tonnes,he gets 160*9.56 = 1529.6 euros

Also given that he gets 75 euros from straws

So in total he gets : 1529.6 + 75 = 1604.4 euros

However his total costs of production of wheat: 1300 euros

So his gain = 1604.4 - 1300 = 304.4 euros