is my calculations correct?

The wizard lives on a secret mountaintop where he brews $3$ types of ale to sell at the local tavern. A liter of his mead sells for $8$ gold profit and requires $5$ bundles of hops and $6$ hours of labor to produce. Beer sells for $6$ gold and requires $4$ bundles of hops and $3$ hours of labor, while barley wine sells for wine right, each liter needs to be stored in its own special barrel, of which the wizard only has $7.$

How many liters of each beverage should the wizard produce this week to gain the most gold if he has $75$ bundles of hops and is willing to work $80$ hours?

Formulation

Let $x1$ be the liters of mead produced.

Let $x2$ be the liters of beer produced.

Let $x3$ be the liters of barley wine produced.

Objective function: $8x1+6x2+10x3$

Constraints

Hopps constraint:

$51+42+3375$

Labour hours constraint:

$61+42+2380$

Barley barrels constraint: $x37$

Primal simplex algorithm and

algorithm and optimal solution

Matricec and mathematical nreliminariec

Calculating the range for a column $$ of a basic variable. $($row $0)$

Let us try to find the range of the $x3$ column. To do this we will first need to find the range of $C3$ or the amount by which the $x3$ coefficient in the objective function can change before we need to recalculate the optimal solution.

To do this we first need to recalculate $CBVB-1$

Then we recalculate the variables in the objective function row of the optimal tablue to see how much we can change $C3$ before we need to pivot again and change the optimal solution.

Calculating the range for a column of a basic variable. $($row $0)$

We see that the only non basic variable in the objective function row affected by the change of C$3$ will be s$3.$

Therefore, the range by which x$3\text{'}$s coefficient can change in the objective function before we have to recalculate the optimal solution is $-5\mathrm{.}2$