$4-$ Consider the following problem and its final simplex tableau to answer the following questions.

"Farmer Jones bakes two types of cake $($chocolate and vanilla$)$ to supplement his income. Each chocolate cake

can be sold for $$1,$ and each vanilla cake can be sold for $25.$ Each chocolate cake requires $20$ minutes of baking time

and uses $4$ eggs. Each vanilla cake requires $40$ minutes of baking time and uses $2$ eggs. Eight hours of baking time and

$30$ eggs are available. Also, Farmer Jones must bake at least $4$ vanilla cakes, because he already has accepted a customer

order. Formulate an LP to maximize Farmer Jones's revenue, then solve the LP$."$

$x_1$: Number of chocolate cakes produced

$x_2$: Number of vanilla cakes produced

maxz$=x_1+\frac{1}{4}{x}_2$

$st.\frac{1}{3}{x}_1+\frac{2}{3}{x}_{2}8(baking$ time capacity$)$

$4x_1+2x_{2}30(egg$ capacity$)$

$x_{2}4(vanilla$ cake demand$)$

$x_1,x_{2}0$

a$)$ If the demand for vanilla.cake increases by one, what will be the change in the optimal objective value?

b$)$ If the available baking time increases by one hour, what will be the change in the optimal objective value?

c$)$ What would be a fair cost to increase the available egg by one?

d$)$ If the number of available eggs increases by one, what will be the change in the optimal objective value