Question

each unit of gadget a requires 2 hours and $20, while each unit of gadget b requires 3 hours and $30.
a) represent the time and budget constraints with inequalities.
b) graph these inequalities on a coordinate plane.
c) identify the feasible region.
d) what is the maximum number of gadgets a and b that can be produced without exceeding the limits?

Explanation:

Step1: Define variables

Let $x$ be the number of Gadget A and $y$ be the number of Gadget B.

Step2: Set up time - constraint inequality

Assume there is a time limit $T$. Since Gadget A requires 2 hours and Gadget B requires 3 hours, the time - constraint is $2x + 3y\leq T$.

Step3: Set up budget - constraint inequality

Assume there is a budget limit $B$. Since Gadget A costs $20$ and Gadget B costs $30$, the budget - constraint is $20x+30y\leq B$. Also, $x\geq0$ and $y\geq0$ as the number of gadgets cannot be negative.

Step4: Graph the inequalities

To graph $2x + 3y\leq T$, first graph the line $2x + 3y=T$ (find the $x$ - intercept by setting $y = 0$: $x=\frac{T}{2}$, and the $y$ - intercept by setting $x = 0$: $y=\frac{T}{3}$). Then shade the region below the line (including the line) since it is $\leq$. Similarly, graph $20x + 30y=B$ ( $x$ - intercept: $x=\frac{B}{20}$, $y$ - intercept: $y=\frac{B}{30}$) and shade the region below it. The non - negativity constraints $x\geq0$ and $y\geq0$ mean we are working in the first quadrant.

Step5: Identify the feasible region

The feasible region is the intersection of the regions defined by $2x + 3y\leq T$, $20x+30y\leq B$, $x\geq0$, and $y\geq0$. It is a convex polygon in the first quadrant.

Step6: Find the maximum number of gadgets

The maximum number of gadgets $N=x + y$ is found by evaluating $N$ at the corner points of the feasible region. Let the corner points be $(x_1,y_1),(x_2,y_2),\cdots$. Calculate $N_1=x_1 + y_1,N_2=x_2 + y_2,\cdots$ and the largest value among them is the maximum number of gadgets that can be produced without exceeding the limits.

However, since the values of $T$ (time limit) and $B$ (budget limit) are not given in the problem, we can only provide the general method.

Answer:

The general method to find the maximum number of gadgets A and B is to follow the steps above. First, set up the inequalities based on time and budget constraints. Then graph them to find the feasible region. Finally, evaluate the objective function $N=x + y$ at the corner - points of the feasible region to get the maximum value. Without specific values for the time and budget limits, a numerical answer cannot be provided.