A community food bank is preparing two types of relief packages: standard boxes and family… — SAT

A community food bank is preparing two types of relief packages: standard boxes and family boxes. Let $x$ represent the number of standard boxes and $y$ represent the number of family boxes. The food bank needs to prepare at least 120 boxes in total. Each standard box contains 3 cans of soup and 2 packages of grain. Each family box contains 6 cans of soup and 5 packages of grain. The food bank has a maximum of 600 cans of soup and a maximum of 450 packages of grain available. Which of the following systems of inequalities represents this situation?

A
$\begin{aligned} x + y &\le 120 \\ 3x + 6y &\ge 600 \\ 2x + 5y &\ge 450 \end{aligned}$
B
$\begin{aligned} x + y &\ge 120 \\ 6x + 3y &\le 600 \\ 5x + 2y &\le 450 \end{aligned}$
$\begin{aligned} x + y &\ge 120 \\ 3x + 6y &\le 600 \\ 2x + 5y &\le 450 \end{aligned}$ Answer
D
$\begin{aligned} x + y &\ge 120 \\ 3x + 2y &\le 600 \\ 6x + 5y &\le 450 \end{aligned}$

Answer

The system of inequalities with

x + y \ge 120

3x + 6y \le 600

, and

2x + 5y \le 450

The correct system of inequalities translates the constraint of preparing at least 120 boxes as

x + y \ge 120

. The maximum limit of 600 cans of soup translates as

3x + 6y \le 600

, since each standard box contains 3 cans and each family box contains 6. The maximum limit of 450 packages of grain translates as

2x + 5y \le 450

, since each standard box contains 2 packages and each family box contains 5.

Step-by-Step Solution

Translate the total package constraint.

x + y \ge 120

The phrase 'at least 120 boxes in total' indicates that the sum of

x

and

y

must be greater than or equal to 120.

Translate the soup constraint.

3x + 6y \le 600

Since each standard box (

x

) has 3 cans of soup and each family box (

y

) has 6 cans of soup, the total soup used is

3x + 6y

. The food bank has 'a maximum of 600 cans,' so this total must be less than or equal to 600.

Translate the grain constraint.

2x + 5y \le 450

Since each standard box (

x

) has 2 packages of grain and each family box (

y

) has 5 packages of grain, the total grain used is

2x + 5y

. The food bank has 'a maximum of 450 packages,' so this total must be less than or equal to 450.

Key Concept

Systems of Linear Inequalities in Two Variables

Estimated Time:1m 30s

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Step-by-Step Solution

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