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Sagot :
Alright, let's break this problem down step by step to determine the correct inequality and draw its graph.
1. Define Variables:
- Let [tex]\( x \)[/tex] represent the number of acres planted with corn.
- Let [tex]\( y \)[/tex] represent the number of acres planted with soybeans.
2. Cost per Acre:
- The cost to plant one acre of corn is \[tex]$240. - The cost to plant one acre of soybeans is \$[/tex]90.
3. Total Cost Constraint:
- The farmer wants to spend no more than \[tex]$7200 on seeds. This gives us the cost constraint. 4. Formulating the Inequality: - The total cost for planting \( x \) acres of corn and \( y \) acres of soybeans can be represented by \( 240x + 90y \). - Since the farmer does not want to exceed \$[/tex]7200, the inequality representing this constraint is:
[tex]\[ 240x + 90y \leq 7200. \][/tex]
- Additionally, we need to ensure that the areas for both crops are nonnegative:
[tex]\[ x \geq 0 \quad \text{and} \quad y \geq 0. \][/tex]
5. Choose the Correct Inequality:
- The correct inequality is:
[tex]\[ 240x + 90y \leq 7200, \quad x \geq 0, \quad y \geq 0. \][/tex]
- Thereby, the answer is Option A:
[tex]\[ 240x + 90y \leq 7200, \quad x \geq 0, \quad y \geq 0. \][/tex]
6. Graphing the Inequality:
- To graph the inequality, first represent the boundary line [tex]\( 240x + 90y = 7200 \)[/tex].
- Simplify the equation by dividing through by 30:
[tex]\[ 8x + 3y = 240. \][/tex]
- To find the intercepts:
- For the [tex]\( y \)[/tex]-intercept ([tex]\( x = 0 \)[/tex]): [tex]\( 8(0) + 3y = 240 \implies y = 80 \)[/tex].
- For the [tex]\( x \)[/tex]-intercept ([tex]\( y = 0 \)[/tex]): [tex]\( 8x + 3(0) = 240 \implies x = 30 \)[/tex].
- Plot these intercepts: (0, 80) and (30, 0).
- Draw the line connecting these points.
- The region of interest is below this line (since [tex]\( 240x + 90y \leq 7200 \)[/tex]) and within the first quadrant (where [tex]\( x \geq 0 \)[/tex] and [tex]\( y \geq 0 \)[/tex]).
Graph:
The region that satisfies all conditions lies within the quadrilateral formed by the x-axis, y-axis, and the line [tex]\( 8x + 3y = 240 \)[/tex]. Below this line and within the first quadrant.
Thus, the farmer should plant acres of corn and soybeans such that the combination of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] falls within this region.
1. Define Variables:
- Let [tex]\( x \)[/tex] represent the number of acres planted with corn.
- Let [tex]\( y \)[/tex] represent the number of acres planted with soybeans.
2. Cost per Acre:
- The cost to plant one acre of corn is \[tex]$240. - The cost to plant one acre of soybeans is \$[/tex]90.
3. Total Cost Constraint:
- The farmer wants to spend no more than \[tex]$7200 on seeds. This gives us the cost constraint. 4. Formulating the Inequality: - The total cost for planting \( x \) acres of corn and \( y \) acres of soybeans can be represented by \( 240x + 90y \). - Since the farmer does not want to exceed \$[/tex]7200, the inequality representing this constraint is:
[tex]\[ 240x + 90y \leq 7200. \][/tex]
- Additionally, we need to ensure that the areas for both crops are nonnegative:
[tex]\[ x \geq 0 \quad \text{and} \quad y \geq 0. \][/tex]
5. Choose the Correct Inequality:
- The correct inequality is:
[tex]\[ 240x + 90y \leq 7200, \quad x \geq 0, \quad y \geq 0. \][/tex]
- Thereby, the answer is Option A:
[tex]\[ 240x + 90y \leq 7200, \quad x \geq 0, \quad y \geq 0. \][/tex]
6. Graphing the Inequality:
- To graph the inequality, first represent the boundary line [tex]\( 240x + 90y = 7200 \)[/tex].
- Simplify the equation by dividing through by 30:
[tex]\[ 8x + 3y = 240. \][/tex]
- To find the intercepts:
- For the [tex]\( y \)[/tex]-intercept ([tex]\( x = 0 \)[/tex]): [tex]\( 8(0) + 3y = 240 \implies y = 80 \)[/tex].
- For the [tex]\( x \)[/tex]-intercept ([tex]\( y = 0 \)[/tex]): [tex]\( 8x + 3(0) = 240 \implies x = 30 \)[/tex].
- Plot these intercepts: (0, 80) and (30, 0).
- Draw the line connecting these points.
- The region of interest is below this line (since [tex]\( 240x + 90y \leq 7200 \)[/tex]) and within the first quadrant (where [tex]\( x \geq 0 \)[/tex] and [tex]\( y \geq 0 \)[/tex]).
Graph:
The region that satisfies all conditions lies within the quadrilateral formed by the x-axis, y-axis, and the line [tex]\( 8x + 3y = 240 \)[/tex]. Below this line and within the first quadrant.
Thus, the farmer should plant acres of corn and soybeans such that the combination of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] falls within this region.
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