Ask a New Question
Menu

Pattie makes two types of chairs: a lawn chair and a living room chair. It takes her 3 hours to make a lawn chair (x) and 5 hours to make a living room chair (y).

Part 1:

If Pattie works a maximum of 50 hours a week, write a two-variable linear inequality to describe the number of possible chairs of each type she can make in 1 week.

3 answers

To express the relationship between the hours spent making lawn chairs and living room chairs, we can define the variables as follows:

  • Let \( x \) represent the number of lawn chairs Pattie makes.
  • Let \( y \) represent the number of living room chairs Pattie makes.

The time it takes for Pattie to make these chairs is as follows:

  • It takes Pattie 3 hours to make one lawn chair, so for \( x \) lawn chairs, the total time is \( 3x \) hours.
  • It takes Pattie 5 hours to make one living room chair, so for \( y \) living room chairs, the total time is \( 5y \) hours.

Since Pattie works a maximum of 50 hours a week, we can write the inequality that represents the total hours she can work as:

\[ 3x + 5y \leq 50 \]

This inequality describes the number of lawn chairs (x) and living room chairs (y) that Pattie can make within a week, given her time constraint.

What combination of lawn and living room chairs is possible for Pattie to make in 1 week based her working a maximum of 50 hours?
Combinations:
10 lawn chairs and 5 living room chairs
3 lawn chairs and 8 living room chairs
5 lawn chairs and 5 living room chairs

To determine if the combinations of lawn chairs and living room chairs are possible for Pattie given her maximum of 50 working hours, we can analyze each combination using the inequality:

\[ 3x + 5y \leq 50 \]

where \( x \) is the number of lawn chairs and \( y \) is the number of living room chairs.

Combination 1: 10 lawn chairs and 5 living room chairs

  • \( x = 10 \)
  • \( y = 5 \)

Calculating the total hours: \[ 3(10) + 5(5) = 30 + 25 = 55 \]

Since \( 55 \) hours \( > 50 \), this combination is not possible.

Combination 2: 3 lawn chairs and 8 living room chairs

  • \( x = 3 \)
  • \( y = 8 \)

Calculating the total hours: \[ 3(3) + 5(8) = 9 + 40 = 49 \]

Since \( 49 \) hours \( \leq 50 \), this combination is possible.

Combination 3: 5 lawn chairs and 5 living room chairs

  • \( x = 5 \)
  • \( y = 5 \)

Calculating the total hours: \[ 3(5) + 5(5) = 15 + 25 = 40 \]

Since \( 40 \) hours \( \leq 50 \), this combination is also possible.

Summary of Combinations:

  1. 10 lawn chairs and 5 living room chairs: Not possible (55 hours)
  2. 3 lawn chairs and 8 living room chairs: Possible (49 hours)
  3. 5 lawn chairs and 5 living room chairs: Possible (40 hours)