Question

question 1 a bakery makes two types of cakes: chocolate and vanilla. each chocolate cake requires 2 hours of labor, and each vanilla cake requires 3 hours of labor. the bakery has at most 80 hours of labor available per week. in addition, each cake (chocolate or vanilla) requires 1 hour in the oven, and the capacity for baking the cakes is 30 oven - hours per week. if x is the number of chocolate cakes produced and y is the number of vanilla cakes, write a system of linear inequalities to model these constraints. inequality for the labor: ____ x + __ y ≤ __ inequality for the oven capacity: __ x + __ y ≤ ____

Explanation:

Step1: Define labor constraint

Each chocolate cake needs 2 labor hours, vanilla needs 3, max 80 hours total.
$2x + 3y \leq 80$

Step2: Define oven constraint

Each cake needs 1 oven hour, max 30 hours total.
$1x + 1y \leq 30$

Step3: Non-negativity constraints (implied)

You can't make negative cakes.
$x \geq 0, \quad y \geq 0$

Answer:

Inequality for the labor: $\boldsymbol{2}x+\boldsymbol{3}y \leq \boldsymbol{80}$
Inequality for the oven capacity: $\boldsymbol{1}x+\boldsymbol{1}y \leq \boldsymbol{30}$
Additional implied constraints: $x \geq 0$, $y \geq 0$