Problem 82 - Algebra and Trigonometry

Question

A patient is not allowed to have more than 330 milligrams of cholesterol per day from a diet of eggs and meat. Each egg provides 165 milligrams of cholesterol. Each ounce of meat provides 110 milligrams. a. Write an inequality that describes the patient's dietary restrictions for \(x\) eggs and \(y\) ounces of meat. b. Graph the inequality. Because \(x\) and \(y\) must be positive, limit the graph to quadrant I only. c. Select an ordered pair satisfying the inequality. What are its coordinates and what do they represent in this situation?

Step-by-Step Solution

Answer

The inequality is \(165x + 110y ≤ 330\). A possible satisfying solution may be (1,1), representing 1 egg and 1 ounce of meat per day.

Step 1: Formulate the inequality

To find an equation that represents the patient's dietary restrictions, we consider the food's cholesterol contributions. Each egg contributes 165 milligrams of cholesterol, which can be expressed as \(165x\). Similarly, each ounce of meat contributes 110 milligrams of cholesterol, which can be expressed as \(110y\). As a result, the sum of these two quantities should not exceed 330 milligrams, resulting in the inequality \(165x + 110y ≤ 330\).

Step 2: Graph the inequality

To graph the inequality, the equation \(y = -1.5x + 3\) (acquired by transforming \(165x + 110y ≤ 330\) into \(y = -\frac{165}{110}x + \frac{330}{110}\)) can be plotted in the positive quadrant of a graph (Quadrant I), as the cholesterol from eggs (\(x\)) and meat (\(y\)) are limited to positive values only. The region bounded by this line and the axes (below the line in Quadrant I) represents all possible solutions for \(x\) and \(y\).

Step 3: Choose an ordered pair

An ordered pair satisfying the inequality from the graph could be, for instance, (1,1). The coordinates represent 1 egg (\(x=1\)) and 1 ounce of meat (\(y=1\)), implying that consuming 1 egg and 1 ounce of meat meets the patient's dietary restrictions as this combination provides less than the 330 milligrams of cholesterol limit per day.