Smash Company has 5000 machine hours available annually to manufacture badminton racquets.The following information is available for the two different racquets produced by Smash: $$\begin{array} { | l | l | }
\hline \text { Pro } & \
\hline \text { Unit sales price } & \$ 300 \
\hline \text { Unit variable costs } & \$ 100 \
\hline \text { Annual demand } & 1900 \text { units } \
\hline \text { Machine time } & 1.25 \text { hours per unit } \
\hline \text { Mid } & \
\hline \text { Unit sales price } & \$ 125 \
\hline \text { Unit variable costs } & \$ 80 \
\hline \text { Annual demand } & 5000 \text { units } \
\hline \text { Machine time } & 4 \text { hours per unit } \
\hline
\end{array}$$ How many units of each racquet should be manufactured for the company to maximize its operating income? (Round your answer to the nearest whole unit.)

Question

Smash Company has 5000 machine hours available annually to manufacture badminton racquets.The following information is available for the two different racquets produced by Smash: $$\begin{array} { | l | l | } 
\hline \text { Pro } & \
\hline \text { Unit sales price } & \$ 300 \
\hline \text { Unit variable costs } & \$ 100 \
\hline \text { Annual demand } & 1900 \text { units } \
\hline \text { Machine time } & 1.25 \text { hours per unit } \
\hline \text { Mid } & \
 \hline \text { Unit sales price } & \$ 125 \
\hline \text { Unit variable costs } & \$ 80 \
\hline \text { Annual demand } & 5000 \text { units } \
\hline \text { Machine time } & 4 \text { hours per unit } \
\hline
\end{array}$$ How many units of each racquet should be manufactured for the company to maximize its operating income? (Round your answer to the nearest whole unit.)

Quizplus · Accepted Answer

To maximize operating income, we need to use the available machine hours efficiently. Let's calculate the machine hours required to produce each racquet:

- Pro: 2.5 machine hours per unit
- Mid: 1.5 machine hours per unit

We can set up the following equation to represent the available machine hours:

2.5P + 1.5M ≤ 5000

where P is the number of Pro racquets and M is the number of Mid racquets. We want to maximize operating income, which is given by:

Operating income = (unit selling price × units sold) - (unit variable cost × units sold) - fixed costs

Let's assume that the unit selling price and unit variable cost are the same for both racquets. We can simplify the equation to:

Operating income = (unit selling price - unit variable cost) × units sold - fixed costs

Since the unit selling price and unit variable cost are constant, we can focus on maximizing units sold. We can set up the following equation to represent the total units sold:

Total units sold = P + M

We want to maximize total units sold subject to the machine hour constraint. We can use a spreadsheet or a graphing calculator to find the optimal solution.

By graphing the inequality 2.5P + 1.5M ≤ 5000, we can see that the feasible region is a triangle bounded by the x-axis, y-axis, and the line 2.5P + 1.5M = 5000. The corner points of the feasible region are:

(0, 3333.33), (1000, 2666.67), and (1900, 656.67)

We can calculate the total units sold and operating income for each corner point:

(0, 3333.33): Total units sold = 3333, Operating income = $0
(1000, 2666.67): Total units sold = 3666, Operating income = $35,500
(1900, 656.67): Total units sold = 2556, Operating income = $57,700

The optimal solution is to produce 1900 units of Pro and 656 units of Mid, which results in a total units sold of 2556 and an operating income of $57,700. Therefore, the answer is B.

Smash Company Has 5000 Machine Hours Available Annually to Manufacture