Problem $3($Multi$-$Variable Unconstrained Optimization$)$

For the following problems, $(1)$ identify the extrema of the objective

function and $(2)$ characterize each extrema as either a minima, maxima, or

neither.

a$.$ A local farm is looking to maximize the profit of their crops. They are

deciding how much to spend on labor costs $(x)$ and fertilizer costs $(y)$

for the upcoming year. The profit per acre of farm is given by the

following function:

$f(x,y)=20x+25y+6xy-2x^2-6y^2$

Find the values of $x$ and $y$ that will maximize the profit.

Ans: Maximum at $(x=\frac{65}{2},y=\frac{55}{3})$

b$.$ Consider the following two$-$bar truss in the figure below:

The potential energy in this system is given by the following equation:

$f(x_1,x_2)=\frac{EA}{s}(\frac{1}{2s}{)}^2{x}_1^2+\frac{EA}{s}(\frac{h}{s}{)}^2{x}_2^2-P{x}_{1}cos()-P{x}_{2}sin()$Where $E=200$GPa is the Young's Modulus, $A={10}^{-5}{m}^2$ is the cross$-$

sectional area, $s=5\mathrm{.}0m$ is the length of each member, $h=4\mathrm{.}0m$ is the

height of the truss, $P=10kN$ is the applied load, and $=30$ is the angle

at which the load is applied. Under this configuration, the point at the

bottom of the truss will displace by an amount of $x_1$ in the horizontal

direction and $x_2$ in the vertical direction. Find the values of $x_1$ and $x_2$ that

minimize the potential energy of this truss structure.

Ans: Minimum at $(x_1=1\mathrm{.}08,x_2=0\mathrm{.}00977)$

c$.$ Consider the following model for the suspension system of a car,

which utilizes two plates and three springs to absorb a compressive

load from below:

The values of the three spring constants are $k_1=4000\frac{N}{m},k_2=$

$6000\frac{N}{m},$ and $k_3=3000\frac{N}{m}.$ The compressive load has a value of $P=$

$350$ N $.$ Positive values for the vertical displacement of the two plates are $y_1$

and $y_2$ and their positive directions are in the downward direction.

Compute the equilibrium values of the plate displacements in this

configuration.

Ans: $y_1=-0\mathrm{.}035m$ and $y_2=-0\mathrm{.}1517m$