Questions: A manufacturing company makes two types of water skis, a trick ski and a slalom ski. The trick ski requires 9 labor-hours for fabricating and 1 labor-hour for finishing. The slalom ski requires 5 labor-hours for fabricating and 1 labor-hour for finishing. The maximum labor-hours available per day for fabricating and finishing are 135 and 20, respectively. Find the set of feasible solutions graphically for the number of each type of ski that can be produced. If x is the number of trick skis and y is the number of slalom skis produced per day, write a system of linear inequalities that indicates appropriate restraints on x and y. Write an inequality for the constraint on fabricating time. Complete the inequality below. 9x+5y ≤ 135 Write an inequality for the constraint on finishing time. Complete the inequality below. x+y ≤ 20 Are any other inequalities needed? A. Yes, x ≤ 0 and y ≤ 0 B. Yes, x ≥ 0 and y ≥ 0 C. Yes, x ≥ y D. No

A manufacturing company makes two types of water skis, a trick ski and a slalom ski. The trick ski requires 9 labor-hours for fabricating and 1 labor-hour for finishing. The slalom ski requires 5 labor-hours for fabricating and 1 labor-hour for finishing. The maximum labor-hours available per day for fabricating and finishing are 135 and 20, respectively. Find the set of feasible solutions graphically for the number of each type of ski that can be produced. If x is the number of trick skis and y is the number of slalom skis produced per day, write a system of linear inequalities that indicates appropriate restraints on x and y. Write an inequality for the constraint on fabricating time. Complete the inequality below. 9x+5y ≤ 135 Write an inequality for the constraint on finishing time. Complete the inequality below. x+y ≤ 20 Are any other inequalities needed? A. Yes, x ≤ 0 and y ≤ 0 B. Yes, x ≥ 0 and y ≥ 0 C. Yes, x ≥ y D. No
Transcript text: A manufacturing company makes two types of water skis, a trick ski and a slalom ski. The trick ski requires 9 labor-hours for fabricating and 1 labor-hour for finishing. The slalom ski requires 5 labor-hours for fabricating and 1 labor-hour for finishing. The maximum labor-hours available per day for fabricating and finishing are 135 and 20, respectively. Find the set of feasible solutions graphically for the number of each type of ski that can be produced. If $x$ is the number of trick skis and $y$ is the number of slalom skis produced per day, write a system of linear inequalities that indicates appropriate restraints on $x$ and $y$. Write an inequality for the constraint on fabricating time. Complete the inequality below. \[ 9 x+5 y \leq 135 \] Write an inequality for the constraint on finishing time. Complete the inequality below. \[ x+y \leq 20 \] Are any other inequalities needed? A. Yes, $x \leq 0$ and $y \leq 0$ B. Yes, $x \geq 0$ and $y \geq 0$ C. Yes, $x \geq y$ D. No
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Solution

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Solution Steps

To solve this problem, we need to:

  1. Write the system of linear inequalities based on the constraints given.
  2. Identify the feasible region graphically by plotting these inequalities.
  3. Determine if any additional inequalities are needed to define the feasible region.
Step 1: Define Variables and Constraints

Let \( x \) be the number of trick skis produced per day, and \( y \) be the number of slalom skis produced per day.

Step 2: Write Inequality for Fabricating Time

The trick ski requires 9 labor-hours for fabricating, and the slalom ski requires 5 labor-hours for fabricating. The maximum labor-hours available per day for fabricating is 135. Therefore, the inequality for fabricating time is: \[ 9x + 5y \leq 135 \]

Step 3: Write Inequality for Finishing Time

Both the trick ski and the slalom ski require 1 labor-hour for finishing. The maximum labor-hours available per day for finishing is 20. Therefore, the inequality for finishing time is: \[ x + y \leq 20 \]

Step 4: Non-Negativity Constraints

Since the number of skis produced cannot be negative, we need the following inequalities: \[ x \geq 0 \quad \text{and} \quad y \geq 0 \]

Step 5: Determine Additional Inequalities

The question asks if any other inequalities are needed. The options are:

  • A. Yes, \( x \leq 0 \) and \( y \leq 0 \)
  • B. Yes, \( x \geq 0 \) and \( y \geq 0 \)
  • C. Yes, \( x \geq y \)
  • D. No

From the non-negativity constraints, we know that \( x \geq 0 \) and \( y \geq 0 \) are necessary. Therefore, the correct answer is: \[ \text{B. Yes, } x \geq 0 \text{ and } y \geq 0 \]

Final Answer

\[ \boxed{\text{B. Yes, } x \geq 0 \text{ and } y \geq 0} \]

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