Q: A farmer wants to plant corn and wheat. He wants to plant at least 30 acres of corn and at least 20 acres of wheat, but no more than 60 acres in total. If $x$ is the number of acres of corn and $y$ is the number of acres of wheat, which system of inequalities represents these constraints?

$\begin{cases} x \geq 30 \\ y \geq 20 \\ x + y \leq 60 \end{cases}$

Question 1

A company produces bicycles and wagons. Each bicycle requires 4 hours to make and each wagon requires 6 hours. The company has at most 60 hours available. If $x$ is the number of bicycles and $y$ is the number of wagons, which inequality best represents this constraint?

Accepted Answer

$4x + 6y \leq 60$

Question 2

A farmer wants to plant corn and wheat. He wants to plant at least 30 acres of corn and at least 20 acres of wheat, but no more than 60 acres in total. If $x$ is the number of acres of corn and $y$ is the number of acres of wheat, which system of inequalities represents these constraints?

Accepted Answer

$\begin{cases} x \geq 30 \ y \geq 20 \ x + y \leq 60 \end{cases}$

Question 3

A toy company makes small and large wagons. Each small wagon requires 4 hours to make and each large wagon requires 6 hours. The company has at most 60 hours available and must make at least 5 small wagons. If $x$ is the number of small wagons and $y$ is the number of large wagons, which of the following is NOT a correct constraint?

Accepted Answer

$4x + 6y \geq 60$

Question 4

A company wants to maximize profit by producing $x$ bicycles and $y$ wagons. The profit function is $P = 30x + 10y$. If the feasible region is bounded by the constraints $x \geq 0$, $y \geq 0$, $x + y \leq 50$, and $2x + y \leq 80$, what is the maximum profit?

Accepted Answer

$P = 900$

Question 5

Given the system of inequalities $x + y \leq 21$, $2x + y \leq 55$, $y \leq 3$, and $x \geq 0$, what is the maximum value of the objective function $P = 4x + 7y$?

Accepted Answer

$P = 37$

Question 6

A clothing company produces shirts and dresses. Each shirt requires 8 hours of labor and each dress requires 15 hours. The company can use at most 450 hours of labor. If $x$ is the number of shirts and $y$ is the number of dresses, which inequality represents this constraint?

Accepted Answer

$8x + 15y \leq 450$

Question 7

A company wants to maximize the storage of files using two types of cabinets: Cabinet X and Cabinet Y. Cabinet X costs $10 per unit and Cabinet Y costs $20 per unit. If $x$ is the number of Cabinet X and $y$ is the number of Cabinet Y, which of the following is the best objective function to maximize the storage of files if Cabinet X stores 7 files and Cabinet Y stores 12 files?

Accepted Answer

$	ext{Max } S = 7x + 12y$

Question 8

A company must decide whether to produce 50 bicycles or 50 wagons. If the constraints are $x + y \leq 50$ and $2x + y \leq 80$, which is a feasible decision?

Accepted Answer

50 wagons

Question 9

A farmer wants to maximize profit from planting corn and wheat. Corn yields a profit of $357$ per acre and wheat yields $159$ per acre. If $x$ is the number of acres of corn and $y$ is the number of acres of wheat, what is the correct objective function to maximize profit?

Accepted Answer

$P = 357x + 159y$

Question 10

A toy company makes small and large wagons. Each small wagon requires 4 hours to make and each large wagon requires 6 hours. The company has at most 60 hours available and must make at least 5 small wagons. If $x$ is the number of small wagons and $y$ is the number of large wagons, which of the following is a correct constraint?

Accepted Answer

$x \geq 5$

Question 11

A company produces two products, X and Y. The profit function is $P = 385x + 425y$. If the constraints are $14x + 27y \leq 80$ and $28x + 54y \leq 175$, what is the maximum profit if $x$ and $y$ are integers?

Accepted Answer

$P = 385x + 425y$ at the feasible point that maximizes P$

Question 12

A company wants to maximize profit by producing $x$ bicycles and $y$ wagons. The profit function is $P = 5x + 50y$. If the feasible region is bounded by the constraints $x \geq 0$, $y \geq 0$, $x + y \leq 50$, and $2x + y \leq 80$, what is the maximum profit?

Accepted Answer

$P = 1000$

Question 13

A company produces two products, A and B. Each unit of A requires 6 hours and each unit of B requires 8 hours. The company has at most 72 hours available. Which inequality represents this constraint?

Accepted Answer

$6x + 8y \leq 72$

Question 14

A company wants to maximize profit by producing $x$ small wagons and $y$ large wagons. The profit function is $P = 12x + 20y$. If the constraints are $4x + 6y \leq 60$, $x \geq 5$, $y \geq 0$, what is the maximum profit?

Accepted Answer

$P = 12x + 20y$ at the feasible point that maximizes P$

Question 15

A clothing company produces shirts and dresses. Each shirt requires 8 hours of labor and each dress requires 15 hours. The company can use at most 450 hours of labor. If $x$ is the number of shirts and $y$ is the number of dresses, which of the following is NOT a correct constraint?

Accepted Answer

$8x + 15y \geq 450$

Question 16

A company wants to maximize profit by producing $x$ units of product X and $y$ units of product Y. The profit function is $P = 45x + 65y$. If the constraints are $x + y \leq 16$, $18x + 35y \leq 450$, $x \geq 0$, $y \geq 0$, what is the maximum profit?

Accepted Answer

$P = 45x + 65y$ at the feasible point that maximizes P$

Question 17

A company wants to maximize profit by producing $x$ units of product X and $y$ units of product Y. The profit function is $P = 385x + 425y$. If the constraints are $14x + 27y \leq 80$ and $28x + 54y \leq 175$, which of the following is a correct constraint?

Accepted Answer

$14x + 27y \leq 80$

Question 18

A company wants to maximize profit by producing $x$ units of product X and $y$ units of product Y. The profit function is $P = 385x + 425y$. If the constraints are $14x + 27y \leq 80$ and $28x + 54y \leq 175$, which of the following is NOT a correct constraint?

Accepted Answer

$14x + 27y \geq 80$

Linear Programming and Systems of Inequalities in College Algebra

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