IDNLearn.com offers a comprehensive solution for all your question and answer needs. Find accurate and detailed answers to your questions from our experienced and dedicated community members.
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.
Your participation means a lot to us. Keep sharing information and solutions. This community grows thanks to the amazing contributions from members like you. Your questions deserve accurate answers. Thank you for visiting IDNLearn.com, and see you again for more solutions.
