Answered

At Westonci.ca, we connect you with experts who provide detailed answers to your most pressing questions. Start exploring now! Get immediate answers to your questions from a wide network of experienced professionals on our Q&A platform. Get quick and reliable solutions to your questions from a community of experienced experts on our platform.

Seed costs for a farmer are [tex]$\$[/tex] 240[tex]$ per acre for corn and $[/tex]\[tex]$ 90$[/tex] per acre for soybeans. How many acres of each crop should the farmer plant if she wants to spend no more than [tex]$\$[/tex] 7200$ on seed? Express your answer as a linear inequality with appropriate nonnegative restrictions and draw its graph.

Let [tex] x [/tex] be the number of acres planted with corn and let [tex] y [/tex] be the number of acres planted with soybeans. Choose the correct inequality below:

A. [tex] 240x + 90y \leq 7200, x \geq 0, y \geq 0 [/tex]
B. [tex] 240x + 90y \geq 7200, x \geq 0, y \geq 0 [/tex]
C. [tex] 240x + 90y \ \textgreater \ 7200, x \geq 0, y \geq 0 [/tex]
D. [tex] 240x + 90y \ \textless \ 7200, x \geq 0, y \geq 0 [/tex]

Use the graphing tool to graph the inequality and the boundary lines representing the nonnegative constraints.

Sagot :

GinnyAnswer
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.
We appreciate your time. Please come back anytime for the latest information and answers to your questions. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.