Questions: A manufacturing company makes two types of water skis, a trick ski and a slalom ski. The relevant manufacturing data are given in the table. Department Labor-Hours per Ski Maximum Labor-Hours Available per Day --------- Trick Ski Slalom Ski Fabricating 9 6 216 Finishing 1 1 27 Answer parts (A), (B), and (C) below. (A) If the profit on a trick ski is 30 and the profit on a slalom ski is 50, how many of each type of ski should be manufactured each day to realize a maximum profit? What is the maximum profit? The maximum profit is The maximum occurs when trick skis and slalom skis are produced.

A manufacturing company makes two types of water skis, a trick ski and a slalom ski. The relevant manufacturing data are given in the table.

Department  Labor-Hours per Ski  Maximum Labor-Hours Available per Day
---------
  Trick Ski  Slalom Ski  
Fabricating  9  6  216
Finishing  1  1  27

Answer parts (A), (B), and (C) below.
(A) If the profit on a trick ski is 30 and the profit on a slalom ski is 50, how many of each type of ski should be manufactured each day to realize a maximum profit? What is the maximum profit?

The maximum profit is  The maximum occurs when trick skis and slalom skis are produced.
Transcript text: A manufacturing company makes two types of water skis, a trick ski and a slalom ski. The relevant manufacturing data are given in the table. \begin{tabular}{|c|c|c|c|} \hline \multirow[b]{2}{*}{Department} & \multicolumn{2}{|c|}{Labor-Hours per Ski} & \multirow[t]{2}{*}{Maximum Labor-Hours Available per Day} \\ \hline & Trick Ski & Slalom Ski & \\ \hline Fabricating & 9 & 6 & 216 \\ \hline Finishing & 1 & 1 & 27 \\ \hline \end{tabular} Answer parts (A), (B), and (C) below. (A) If the profit on a trick ski is $\$ 30$ and the profit on a slalom ski is $\$ 50$, how many of each type of ski should be manufactured each day to realize a maximum profit? What is the maximum profit? The maximum profit is $\$ \square$ $\square$ The maximum occurs when $\square$ trick skis and $\square$ $\square$ slalom skis are produced.
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Solution

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Solution Steps

Step 1: Formulate the LP problem

Maximize $P = 30x_1 + 50x_2$ subject to:

  • Fabricating department: $9x_1 + 6x_2 \leq 216$
  • Finishing department: $1x_1 + 1x_2 \leq 27$
  • Non-negativity: $x_1, x_2 \geq 0$
Step 2: Solve the LP problem

Using a linear programming solver, we find the optimal solution:

  • $x_1^* = 0$
  • $x_2^* = 27$
  • Maximum Profit: $1350$

Final Answer:

The optimal production quantities are 0 units of product 1 and 27 units of product 2, yielding a maximum profit of $1350.

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