Problem 82 - College Algebra Essentials

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

a) The inequality is \(165x + 110y \leq 330\).\n b) The graph is a solid line with slope -3/2 where \(x\) and \(y\) are positive. The feasible region is below and including the line in quadrant I. \n c) A feasible solution could be \((1, 1)\), it means the patient can have one egg and one ounce of meat to stay within the dietary restrictions.

Step 1: Formulate the Inequality

Let \(x\) denote the number of eggs and \(y\) denote the ounces of meat. Each egg (x) provides 165 milligrams of cholesterol and each ounce of meat (y) provides 110 milligrams. Since the total cholesterol per day should not exceed 330 milligrams, the inequality can be written as \(165x+110y \leq 330\).

Step 2: Graph the Inequality

To graph the inequality, first, plot the graph for the equality \(165x + 110y = 330\). Start by setting \(x = 0\) to find the \(y\)-intercept and set \(y = 0\) to find the \(x\)-intercept. This will give the line dividing the plane into two regions. Since the inequality allows for the sum to be equal to 330 or less, the line itself will be included in the graph (shown by a solid line). Because both \(x\) and \(y\) must be positive, the feasible region is in quadrant I (the region below and including the line).

Step 3: Find and interpret a feasible solution

A feasible solution is an ordered pair \((x, y)\) that satisfies the inequality \(165x + 110y \leq 330\). For instance, \((1, 1)\) is a solution as substituting \(x = 1\) and \(y = 1\) into the inequality results in \(275 \leq 330\), which is true. In this context, this means consuming one egg and one ounce of meat will give a total of 275 milligrams of cholesterol, which is within the allowed limit of 330 milligrams.