91Ó°ÊÓ

Let the $packageproducebyFriskyPuppermonth=F.$The package contains 1 pound of cereal and cost $1\$/pound$and 1.5 pound of meat and cost $2\$/pound.$

So,we will achieve maximum profit if we have:$1.6F+1.4H$

Here,

On solving above conditions in the question, maximum profit is:$22.2×{10}^4$

So total production cost include cost of production of cereal and meat and package cost($1.40\$$). This givelocalid="1658924309919" $\mathrm{total}\mathrm{cost}\mathrm{of}\mathrm{production}\mathrm{of}\mathrm{a}\mathrm{package}=1+2×1.5+1.40=5.4\$.$.

Hence,localid="1658924239764" $\mathrm{profit}\mathrm{obtain}\mathrm{in}\mathrm{selling}\mathrm{a}\mathrm{package}=\mathrm{Selling}\mathrm{price}-\mathrm{Cost}\mathrm{of}\mathrm{production}.$

$Profit=7-5.4=1.6\$$

So,$maximumprofitofF=1.6F$

Let $\mathrm{the}\mathrm{package}\mathrm{produce}\mathrm{by}\mathrm{Husky}\mathrm{Hound}\mathrm{per}\mathrm{month}=\mathrm{H}$.The package contain 2 pound of cereal and cost $1\$/pound$and 1 pound of meat and costlocalid="1658981373747" $2\$/pound.$So total production cost include cost of production of cereal and meat and package cost $(0.6\$)$.This give localid="1658981380385" $\mathrm{total}\mathrm{cost}\mathrm{of}\mathrm{production}\mathrm{of}\mathrm{a}\mathrm{package}=2×1+2+0.6=4.6\$.$Hence, $profitobtaininsellingapackage=Sellingprice-Costofproduction.$$1.6F+1.4H$.

$Profit=6-4.6=1.4\$$

So, max profit$ofH=1.4H$

Therefore, max profitlocalid="1658923761946" $=1.6F+1.4H$

This is our optimal goal to achieve.

Total cereal available in a month $F+2H≤240000$pounds. Also, Frisky Pup and Husky Hound contain 1 pound and 2 pound cereal respectively.

So, the constraint is: $F+2H≤240000$

Total meat available in a month$H≤180000$pounds. Also, Frisky Pup and Husky Hound contain $1.5$ pound and 1 pound meat respectively. So,

the constraint is: $H≤180000$.

Frisky Pug can produce at the most localid="1658982614215" $F≤110000$package.

This can be expressed as:$F≤110000$ .

And the package quantity can never be negative.

Thus, our required linear program is:

$$\begin{gathered} \mathrm{Maximum}: \\ 1.6\mathrm{H}+1.4\mathrm{H} \end{gathered}$$

Constraints

$$\begin{gathered} \left[1\right]F+2H≤240000 \\ \left[2\right]1.5F+H≤180000 \\ \left[3\right]F≤110000 \\ \left[4\right]Fâ‰\yen 0 \\ \left[5\right]Hâ‰\yen 0 \end{gathered}$$

The graph from above constraints is:

In this graph,$\left("\right)$denote power of 4 .

From the graph, it is clear that we will obtain maximum profit from

$(\mathrm{F},\mathrm{H})=(6×{10}^4,9×{10}^4)$

So, our maximum profit will be:

$$\begin{gathered} 1.6F+1.4H=1.6×6×{10}^4+1.4×9×{10}^4 \\ =22.2×{10}^4 \end{gathered}$$