Question

part 1 of 3 (a) determine the number of grill a units and the number of grill b units that should be produced to maximize profit. the manufacturer should produce 400 grill a units and 200 grill b units to maximize profit. part 2 of 3 (b) what is the maximum profit under these constraints? the maximum profit is $60000. alternate answer: the maximum profit is $60,000. part 3 of 3 (c) if the profit on grill a units is $130 and the profit on grill b units is unchanged, how many of each type of grill unit should the manufacturer produce to maximize profit? in this case, the manufacturer should produce grill a units and grill b units.

Explanation:

Step1: Identify given values

We know from part (a) and (b) that 400 grill A units and 200 grill B units give a maximum profit of $60000, and profit per grill A is $130 and profit per grill B is unchanged.

Step2: Analyze profit - unit relationship

Let the profit per grill B be $p$. The profit function is $P = 130\times400 + p\times200=60000$. Solving for $p$:
\[

$$\begin{align*} 130\times400+200p&=60000\\ 52000 + 200p&=60000\\ 200p&=60000 - 52000\\ 200p&=8000\\ p& = 40 \end{align*}$$

\]
The profit - maximization condition remains the same with the new profit per grill A. Let the number of grill A be $x$ and grill B be $y$. The profit function is $Z=130x + 40y$. Assuming the same constraints as before (not given in full here but based on the previous optimal solution). Since the optimal solution structure is likely based on the ratio of profits and resource - usage (not shown but typical in linear programming problems), the optimal production levels remain the same when only the profit of one product changes in a certain way.

Answer:

400 grill A units and 200 grill B units