Solve the problems below that pertain to a farmer planting wheat. (a) If a farme

Question

Solve the problems below that pertain to a farmer planting wheat. (a) If a farmer plants x units of wheat in his field, 0 sxs 100, the yield will be 10x - X_ units. How much wheat should he plant 10 for the maximum yield? 50 units of wheat (b) In the problem above, it costs the farmer $100 for each unit of wheat he plants, and he is able to sell each unit he harvests for $50. How much should he plant to maximize his profit? 51 Xunits of wheat Recall profit equals revenue minus cost. Express the profit in terms of x. Observe the restricted domain. Why are the critical points as well as the endpoints of the restricted domain used for determining any maximum values?

Explanation / Answer

a) We have given y=10x-x2/10,0<=x<=100

To Maximize y,dy/dx=10-2x/10=0

10=x/5 implies x=50

he should plant 50 units of wheat

b) Profit =Revenue function-Cost function

Profit =P=R(x)-C(x)=50(10x-x2/10)-100x =500x-5x2-100x=400x-5x2

P=400x-5x2

To Maximize P,dP/dx=0

dP/dx=0

400-10x=0

10x=400

x=40

to maximize profit P,he should plant 40 units of wheat

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