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Difficulty: MediumSystems of Linear Inequalities in Two Variables

A community garden has a total area of 1,2001,200 square feet available for planting tomatoes and squash. Each tomato plant requires 88 square feet of space, and each squash plant requires 1515 square feet of space. The garden coordinators want to plant at least 100100 plants in total, but they want the number of squash plants to be no more than twice the number of tomato plants. If tt represents the number of tomato plants and ss represents the number of squash plants, which of the following systems of inequalities represents this situation?

  1. A
    {8t+15s≥1,200t+s≤100s≤2t\begin{cases} 8t + 15s \geq 1,200 \\ t + s \leq 100 \\ s \leq 2t \end{cases}
  2. B
    {8t+15s≤1,200t+s≥100t≤2s\begin{cases} 8t + 15s \leq 1,200 \\ t + s \geq 100 \\ t \leq 2s \end{cases}
  3. {8t+15s≤1,200t+s≥100s≤2t\begin{cases} 8t + 15s \leq 1,200 \\ t + s \geq 100 \\ s \leq 2t \end{cases}Answer
  4. D
    {8t+15s≤1,200t+s≥100s≥2t\begin{cases} 8t + 15s \leq 1,200 \\ t + s \geq 100 \\ s \geq 2t \end{cases}

Answer

The system of inequalities containing the constraints 8t+15s≤1,2008t + 15s \leq 1,200, t+s≥100t + s \geq 100, and s≤2ts \leq 2t.
The correct system of inequalities maps the garden's constraints accurately: 8t+15s≤1,2008t + 15s \leq 1,200 limits the space usage to a maximum of 1,2001,200 square feet; t+s≥100t + s \geq 100 ensures there are at least 100100 plants in total; and s≤2ts \leq 2t restricts the number of squash plants to at most twice the number of tomato plants.

Step-by-Step Solution

1
Translate the area constraint.
8t+15s≤1,2008t + 15s \leq 1,200
Since the total area available is 1,2001,200 square feet and each tomato plant requires 88 square feet while each squash plant requires 1515 square feet, the combined space used by tt tomato plants and ss squash plants cannot exceed 1,2001,200.
2
Translate the total plant count constraint.
t+s≥100t + s \geq 100
The coordinators want at least 100100 plants, meaning the sum of the tomato and squash plants must be greater than or equal to 100100.
3
Translate the plant ratio constraint.
s≤2ts \leq 2t
The number of squash plants (ss) must be no more than (less than or equal to) twice the number of tomato plants (2t2t).
4
Combine the inequalities into a single system.
{8t+15s≤1,200t+s≥100s≤2t\begin{cases} 8t + 15s \leq 1,200 \\ t + s \geq 100 \\ s \leq 2t \end{cases}
Grouping these three constraints forms the mathematical model that represents the given scenario.

Key Concept

Translating real-world constraints into a system of linear inequalities in two variables.
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Systems of Linear Inequalities in Two Variables

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