According to a model developed by economists E. Heady and J....

According to a model developed by economists E. Heady and J. Pesek, if fertilizer made from $N$ pounds of nitrogen and $P \mathrm{lb}$ of phosphate is used on an acre of farmland, then the yield of corn (in bushels per acre) is $$ Y=7.5+0.6 N+0.7 P-0.001 N^{2}-0.002 P^{2}+0.001 N P $$ A farmer intends to spend $\$ 30 /$ acre on fertilizer. If nitrogen costs 25 cents $/ \mathrm{lb}$ and phosphate costs 20 cents $/ \mathrm{lb}$, which combination of $\mathrm{N}$ and $P$ produces the highest yield of corn?

According to a model developed by economists E. Heady and
J. Pesek, if fertilizer made from $N$ pounds of nitrogen and $P \mathrm{lb}$ of phosphate is used on an acre of farmland, then the yield of corn (in bushels per acre) is
$$
Y=7.5+0.6 N+0.7 P-0.001 N^{2}-0.002 P^{2}+0.001 N P
$$
A farmer intends to spend $\$ 30 /$ acre on fertilizer. If nitrogen costs 25 cents $/ \mathrm{lb}$ and phosphate costs 20 cents $/ \mathrm{lb}$, which combination of $\mathrm{N}$ and $P$ produces the highest yield of corn?

Key Concepts

Constrained Optimization

Constrained optimization is the process of finding the maximum or minimum of an objective function subject to certain restrictions or constraints. This method is fundamental in fields like economics and engineering, where decision-making often must operate within limited resources or other boundaries.

Quadratic Function Optimization

Optimizing quadratic functions involves finding the vertex of a quadratic equation, which corresponds to the maximum or minimum value of the function. In practice, this is often done when the relationship between variables is modeled by a quadratic expression, enabling a clear identification of optimal conditions.

Lagrange Multipliers

The method of Lagrange multipliers is a powerful tool used to solve optimization problems that include constraints. By introducing additional variables (the multipliers), one can convert a constrained optimization problem into a system of equations, making it easier to identify the points where the objective function is optimized while satisfying the constraints.

Budget Constraints in Optimization

A budget constraint is a specific type of linear constraint that limits the total expenditure on various inputs to a fixed amount. Incorporating budget constraints is common in economic models and resource allocation problems to ensure that the optimal solution remains feasible within available financial resources.

Transcript

00:01 Pounds of nitrogen and p is pounds of phosphate.

00:03 And why is the yield in bushels of corn? so how much corn will be produced with respect to how much nitrogen and phosphate you put on the field? this farmer has $30 an acre to spend on nitrogen and phosphate.

00:22 And nitrogen is 25 cents a pound.

00:25 And phosphate is 20 cents a pound.

00:27 Find n &p to maximize z.

00:30 Yield.

00:31 So we want to maximize y.

00:34 There's a bunch of decimals in there.

00:36 So i'm going to maximize a thousand y, which if we find the maximum of a thousand y, it will be the same as finding the maximum of y.

00:46 So that's 7500 plus 600, oh 600 n plus 700 p, minus n squared minus 2 p squared plus np.

01:01 Okay and that's subject to the constraint that he has $30 to spend and cost 25 cents a pound for n and 20 cents a pound for p.

01:27 Okay, so we're going to take the derivative of 1 ,000 y and set it equal to zero and solve.

01:32 But first we've got to get the n or p out of there and it really doesn't matter which one you solve for.

01:38 I'm going to solve for p, i guess.