NGPF 4.8 Constraints and Optimization

Which of the following activities would satisfy all friends' desires and constraints?

You’re deciding what activity to do on a weekend with your friends.

You want to find something that balances everyone's desires around time and money.

Figure out what activities, if any, are options that will make everyone happy.

Friends’ Desires:

• Sandra: Wants to spend at least 3 hours together but has a budget of $50

• Larry: 5 hour limit and doesn’t care how much it costs

• Bethany: Is fine with any length of time, but has a budget of $35

• Chris: Wants to spend at least 2 but no more than 4 hours, and has budgeted $60 for the weekend

Activities available:

• Laser Tag: $30 each and 4 hours total

• Sailing: $20 each and a 6 hour total

• Hang Gliding: $50 and a 3 hour total

• Movie: $20 and 3 hour total

• Go Karting: $15 and 1 hour total

• Hiking: $5 and 5 hour total

• Pottery Class: $45 and 4 hour total

• Laser Tag

• Go Karting

• Movie

• Pottery Class

• Hiking

Which friend has the most restrictive budget for the weekend activity?

You’re deciding what activity to do on a weekend with your friends.

You want to find something that balances everyone's desires around time and money.

Figure out what activities, if any, are options that will make everyone happy.

Friends’ Desires:

• Sandra: Wants to spend at least 3 hours together but has a budget of $50

• Larry: 5 hour limit and doesn’t care how much it costs

• Bethany: Is fine with any length of time, but has a budget of $35

• Chris: Wants to spend at least 2 but no more than 4 hours, and has budgeted $60 for the weekend

Activities available:

• Laser Tag: $30 each and 4 hours total

• Sailing: $20 each and a 6 hour total

• Hang Gliding: $50 and a 3 hour total

• Movie: $20 and 3 hour total

• Go Karting: $15 and 1 hour total

• Hiking: $5 and 5 hour total

• Pottery Class: $45 and 4 hour total

• Sandra

• Larry

• Bethany

• Chris

Which activity has the longest total time commitment?

You’re deciding what activity to do on a weekend with your friends.

You want to find something that balances everyone's desires around time and money.

Figure out what activities, if any, are options that will make everyone happy.

Friends’ Desires:

• Sandra: Wants to spend at least 3 hours together but has a budget of $50

• Larry: 5 hour limit and doesn’t care how much it costs

• Bethany: Is fine with any length of time, but has a budget of $35

• Chris: Wants to spend at least 2 but no more than 4 hours, and has budgeted $60 for the weekend

Activities available:

• Laser Tag: $30 each and 4 hours total

• Sailing: $20 each and a 6 hour total

• Hang Gliding: $50 and a 3 hour total

• Movie: $20 and 3 hour total

• Go Karting: $15 and 1 hour total

• Hiking: $5 and 5 hour total

• Pottery Class: $45 and 4 hour total

• Laser Tag

• Go Karting

• Hiking

• Pottery Class

• Sailing

What is the first step in solving systems with many constraints?

• Graph the constraints

• Eliminate redundant constraints

• List the constraints

• List solutions

• Plot the possible solution points

Identify a point that falls within the shaded solution region on the graph where all the possible regions overlap.

• (4, 3)

• (2, 5)

• (2, 3)

• (3, 5)

• (3, 3)

What is the practice of finding the best of all possible solutions, which involves testing solutions in an equation to either maximize or minimize a certain parameter?

• Constraint Listing

• Redundancy Elimination

• Optimization

• Graphical Analysis

• Budgeting

Answer explanation

Optimization is the practice of finding the BEST of all possible solutions. It involves first finding the set of all possible solutions and second, testing those solutions in an optimization equation that either maximizes or minimizes a certain parameter. Graphing is particularly helpful because possible solutions will ALWAYS be found at/near the corners of a solution region, or in other words, where two or more constraints meet.

In the situation described below, what is the maximum revenue that can be generated from selling widgets and doodads in a 40-hour work week?

You work for a factory that makes widgets and doodads.

Each widget takes 3 hours to make and sells for $20, while doodads take 4 hours each and sell for $30.

You want to make as many widgets(x) and doodads(y) efficiently in a 40 hour work week to maximize your sales revenue.

• $260, by making 13 widgets and 0 doodads

• $300, by making 0 widgets and 10 doodads

• $320, by making 7 widgets and 6 doodads

• $350, by making 4 widgets and 7 doodads

• $360, by making 6 widgets and 8 doodads

What is the constraint equation listed in the following situation for the 40-hour work week?

You work for a factory that makes widgets and doodads.

Each widget takes 3 hours to make and sells for $20, while doodads take 4 hours each and sell for $30.

How many widgets(x) and doodads(y) should you make in a 40 hour work week to maximize your sales revenue?

• 3x + 4y ≤ 40

• 3x + 4y ≥ 40

• 4x - 3y ≤ 40

• 4x - 3y ≥ 40

• 3x + 4y > 40

Answer explanation

Which set of inequalities corresponds to the graph that is shaded below the line and to the left of the y-axis?

• 2x + 3y ≤ 15,

  y ≤ 2,

  y ≥ -4

• 3x + 2y ≤ 15,

  x ≥ 1,

  y ≤ 3

• 4x - 2y ≤ 10,

  y < 3,

  x < 3

• 2x + 3y ≤ 15,

  y ≤ 3,

  x ≥ 1

• 4x - 2y ≥ 10,

  x ≤ 3,

  y ≥ 1

Answer explanation

Given the constraint inequality

4x - 2y ≥ 10

when x is the amount of hours you work

and y is the number of birdhouses made

Is it possible to make $400 by selling birdhouses for $11 each, by working part-time for just 20 hours a week?

• Yes, because you can sell 40 birdhouses in 20 hours and make $440.

• No, because you can only make a maximum of $385 because you can only make 35 bird houses in 20 hours.

• Yes, because you can work extra hours to make more birdhouses.

• No, because the inequality does not allow for making 20 birdhouses.

• No, because there are not enough materials to make 40 birdhouses.