A shipping company uses a cargo plane to transport two types of cargo containers: Type A a… — SAT

A shipping company uses a cargo plane to transport two types of cargo containers: Type A and Type B. Let $x$ represent the number of Type A containers, and let $y$ represent the number of Type B containers. The constraints on the shipment are represented by the system of inequalities below:

\begin{aligned} x + y &\leq 45 \\ 3x + 5y &\leq 165 \end{aligned}

If the plane must carry at least $10$ Type A containers, what is the maximum number of Type B containers the plane can transport?

A
25
27Answer
C
35
D
39

Answer

The correct answer of 27 is found by substituting the minimum value of

x

, which is 10, into the second inequality constraint. This yields

3(10) + 5y \leq 165

, which simplifies to

5y \leq 135

, or

y \leq 27

. Since the coordinate pair (10, 27) also satisfies the first inequality constraint (

10 + 27 = 37 \leq 45

), 27 is the maximum possible value for

y

Step-by-Step Solution

Identify the given constraints and the variable to maximize.

We are given the system of inequalities

x + y \leq 45

and

3x + 5y \leq 165

, with

x \geq 10

, and we want to find the maximum possible value of

y

This establishes the mathematical model and boundaries for the problem.

Express

y

in terms of

x

for both inequalities to see which constraint is more restrictive.

From the first inequality:

y \leq 45 - x

. From the second inequality:

3x + 5y \leq 165 \implies 5y \leq 165 - 3x \implies y \leq 33 - 0.6x

This allows us to analyze

y

as a function of

x

under both constraints.

Substitute the minimum possible value of

x

, which is 10, into both boundary expressions for

y

Under the first constraint:

y \leq 45 - 10 = 35

. Under the second constraint:

y \leq 33 - 0.6(10) = 27

Since both boundary lines have negative slopes, the maximum value of

y

will occur when

x

is at its minimum value of 10.

Determine the most restrictive upper bound for

y

when

x = 10

The value of

y

must satisfy both

y \leq 35

and

y \leq 27

, so the maximum possible value is 27.

A solution to a system of inequalities must satisfy all inequalities in the system simultaneously.

Key Concept

Solving systems of linear inequalities in context by evaluating boundary conditions and identifying constraints.

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