13. A farmer plans to create a rectangular garden that he will enclose with chicken wire. The garden can be no more than 30 ft wide. The farmer would like to use at most 180 ft of... 13. A farmer plans to create a rectangular garden that he will enclose with chicken wire. The garden can be no more than 30 ft wide. The farmer would like to use at most 180 ft of chicken wire. a. Write a system of linear inequalities that models this situation. b. Graph the system to show all possible solutions. Write a system of equations to model each situation. Solve by any method. 14. A writing workshop enrolls novelists and poets in a ratio of 5:3. There are 24 people at the workshop. How many novelists are there? How many poets are there? 15. A chemist has one solution containing 30% insecticide and another solution containing 50% insecticide. How much of each solution should the chemist mix to get 200 L of a 42% insecticide?

Question image

Understand the Problem

The image contains three math problems related to different scenarios: gardening with constraints, education with ratios, and chemistry with mixture percentages. The goal is to apply mathematical concepts to solve each problem, specifically using systems of inequalities, ratios, and mixture calculations.

Answer

13. a. $ w \leq 30 $, $ w + l \leq 90 $, $ w \geq 0 $, $ l \geq 0 $ 14. 15 novelists, 9 poets 15. 80 L of 30% solution, 120 L of 50% solution
Answer for screen readers
  1. a. $ w \leq 30 $ $ w + l \leq 90 $ $ w \geq 0 $ $ l \geq 0 $
  2. 15 novelists and 9 poets
  3. 80 L of the 30% solution and 120 L of the 50% solution

Steps to Solve

  1. Define variables for the garden problem

Let $w$ be the width of the rectangular garden and $l$ be the length.

  1. Write the inequality for the width constraint

The garden can be no more than 30 ft wide, so $w \leq 30$

  1. Write the inequality for the perimeter constraint

The farmer wants to use at most 180 ft of chicken wire (perimeter), so $2w + 2l \leq 180$

  1. Simplify the perimeter inequality

Divide both sides of the inequality by 2, giving $w + l \leq 90$

  1. State the non-negativity constraints

Since width and length cannot be negative, include the constraints $w \geq 0$ and $l \geq 0$

  1. State the system of inequalities

The system of linear inequalities is: $ w \leq 30 $ $ w + l \leq 90 $ $ w \geq 0 $ $ l \geq 0 $

  1. Solve the education problem - define variables Let $n$ be the number of novelists and $p$ be the number of poets.

  2. Write the equation for the total number of people $ n + p = 24 $

  3. Write the equation for the ratio of novelists to poets $ \frac{n}{p} = \frac{5}{3} $

  4. Solve for $n$ in terms of $p$ in the ratio equation $ n = \frac{5}{3}p $

  5. Substitute the expression for $n$ into the total number of people equation $ \frac{5}{3}p + p = 24 $

  6. Solve for $p$ $ \frac{8}{3}p = 24 $ $ p = 24 \cdot \frac{3}{8} = 9 $

  7. Solve for $n$ $ n = 24 - p = 24 - 9 = 15 $

  8. State the answers for novelists and poets There are 15 novelists and 9 poets.

  9. Solve the chemistry problem - define variables Let $x$ be the amount (in liters) of the 30% insecticide solution, and $y$ be the amount (in liters) of the 50% insecticide solution.

  10. Write the equation for the total volume of the mixture $ x + y = 200 $

  11. Write the equation for the amount of insecticide in the mixture $ 0.30x + 0.50y = 0.42(200) $

  12. Simplify the insecticide equation $ 0.30x + 0.50y = 84 $

  13. Solve for $x$ in terms of $y$ in the volume equation $ x = 200 - y $

  14. Substitute the expression for $x$ into the insecticide equation $ 0.30(200 - y) + 0.50y = 84 $

  15. Solve for $y$ $ 60 - 0.30y + 0.50y = 84 $ $ 0.20y = 24 $ $ y = \frac{24}{0.20} = 120 $

  16. Solve for $x$ $ x = 200 - y = 200 - 120 = 80 $

  17. State the answer for the amount of each solution The chemist should mix 80 L of the 30% solution and 120 L of the 50% solution.

  1. a. $ w \leq 30 $ $ w + l \leq 90 $ $ w \geq 0 $ $ l \geq 0 $
  2. 15 novelists and 9 poets
  3. 80 L of the 30% solution and 120 L of the 50% solution

More Information

  1. b. To graph this system, we would graph each inequality on the coordinate plane. The region where all inequalities overlap represents the possible solutions.
  2. The ratio 5:3 means that for every 5 novelists, there are 3 poets.
  3. Mixing solutions involves considering both the volumes and the concentrations to achieve the desired final concentration.

Tips

  1. Forgetting non-negativity constraints (Gardening): A common mistake is to not include $w \geq 0$ and $l \geq 0$. Widths and lengths can't be negative, so these inequalities are necessary.
  2. Incorrectly setting up ratio equations (Education): Setting up the ratio equation incorrectly (e.g., $\frac{n}{p} = \frac{3}{5}$) would lead to the wrong answer.
  3. Incorrectly setting up mixture equations (Chemistry): A common mistake is setting up the insecticide amount equation incorrectly by not multiplying the final concentration by the total volume.

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