A nutritionist is designing a meal plan containing x grams of protein and y grams of carbo… — SAT

A nutritionist is designing a meal plan containing $x$ grams of protein and $y$ grams of carbohydrates. The meal plan must satisfy the following system of inequalities:

\begin{aligned} y &\ge 1.5x + 15 \\ 3x + 2y &\le 120 \end{aligned}

What is the maximum possible number of grams of protein, $x$ , that can be included in the meal plan?

Answer: 15 grams

Answer

The maximum possible number of grams of protein that can be included is 15.

To find the maximum possible value of

x

, we determine the region defined by the system of inequalities. The system restricts the values to the region above the line

y = 1.5x + 15

and below the line

3x + 2y = 120

. Since the first inequality limits

y

from below and the second limits

y

from above, the feasible region narrows as

x

increases, terminating at the intersection of the two boundary lines. Substituting

y = 1.5x + 15

into

3x + 2y = 120

gives

3x + 2(1.5x + 15) = 120

. Simplifying this yields

3x + 3x + 30 = 120

, which simplifies further to

6x = 90

, giving

x = 15

. Thus, the maximum value of

x

is 15.

Step-by-Step Solution

Identify the boundary lines of the system of inequalities.

The boundary lines are

y = 1.5x + 15

and

3x + 2y = 120

The maximum value of

x

under these linear constraints occurs at the intersection of the boundary lines of the feasible region.

Substitute the expression for

y

from the first boundary equation into the second equation.

3x + 2(1.5x + 15) = 120

This allows us to solve for

x

by eliminating

y

Simplify the equation and solve for

x

3x + 3x + 30 = 120 \implies 6x + 30 = 120 \implies 6x = 90 \implies x = 15

Solving the linear equation gives the

x

-coordinate of the intersection point.

Verify that this point lies in the feasible region and represents the maximum possible value of

x

x=15

y=37.5

. Since

y \ge 1.5x + 15

restricts the region above the line and

3x + 2y \le 120

restricts it below the line, the region lies to the left of the intersection point

(15, 37.5)

. Thus, the maximum value of

x

is 15.

Confirming the geometry of the feasible region ensures the intersection point is indeed the maximum value.

Key Concept

Solving systems of linear inequalities to find the boundaries and extreme values of a feasible region.

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