Question

many elevators have a capacity of 3000 pounds. complete parts (a) through (c) below. b. graph the inequality. because x and y must be positive, limit the graph to quadrant i only. use the graphing tool to graph the inequality. c. select an ordered pair that satisfies the inequality. what are the coordinates of the pair? what do the coordinates of the pair represent in the given situation? the point satisfies the inequality. the coordinates represent the fact that when children and adults are in the elevator, the elevator is overloaded.

Explanation:

1. Assume the weights:

• Let's assume the average weight of a child is \(c\) pounds and the average weight of an adult is \(a\) pounds. Let \(x\) be the number of children and \(y\) be the number of adults. The inequality representing the elevator - capacity situation is \(cx + ay\leq3000\). Since we are working in quadrant I (\(x\geq0,y\geq0\)), we first need to rewrite the inequality in slope - intercept form \(y\leq-\frac{c}{a}x+\frac{3000}{a}\). But without specific values of \(c\) and \(a\), we'll work in a general sense for graphing.
• To graph the line \(cx + ay = 3000\) (the boundary line of the inequality), we find the \(x\) - intercept and \(y\) - intercept.
• Find the \(x\) - intercept:
• Set \(y = 0\) in the equation \(cx+ay = 3000\). Then \(cx=3000\), so \(x=\frac{3000}{c}\).
• Find the \(y\) - intercept:
• Set \(x = 0\) in the equation \(cx + ay = 3000\). Then \(ay = 3000\), so \(y=\frac{3000}{a}\).
• Since \(x\geq0,y\geq0\), we draw a solid line (if the inequality is \(\leq\)) or a dashed line (if the inequality is \(<\)) connecting the \(x\) - intercept and \(y\) - intercept and shade the region below the line (because \(y\leq-\frac{c}{a}x+\frac{3000}{a}\)) in quadrant I.

1. Select an ordered - pair:

• Let's assume the average weight of a child \(c = 50\) pounds and the average weight of an adult \(a = 150\) pounds. Then the inequality is \(50x + 150y\leq3000\), which simplifies to \(x + 3y\leq60\).
• We want to find an ordered pair \((x,y)\) that satisfies the inequality and represents an over - loaded situation. An over - loaded situation means \(50x + 150y>3000\) (or \(x + 3y>60\)).
• Let's choose the point \((50,10)\).
• Substitute \(x = 50\) and \(y = 10\) into the left - hand side of the inequality \(50x+150y\):
• \(50\times50+150\times10=2500 + 1500=4000\). Since \(4000>3000\), the point \((50,10)\) represents a situation where the elevator is overloaded.

Step1: Set up the inequality

Let \(x\) be the number of children and \(y\) be the number of adults. With average child weight \(c\) and adult weight \(a\), the capacity inequality is \(cx + ay\leq3000\).

Step2: Graph the boundary line

Find \(x\) - intercept (\(y = 0\)) and \(y\) - intercept (\(x = 0\)) of \(cx+ay = 3000\) and draw the line in quadrant I. Shade the appropriate region for the inequality.

Step3: Select an over - loaded point

Assume \(c = 50\), \(a = 150\). The inequality is \(x + 3y\leq60\). Choose \((x,y)=(50,10)\) and check it in the original non - simplified inequality \(50x + 150y\).

Answer:

The point \((50,10)\) satisfies the over - loading condition. The coordinates represent that when 50 children and 10 adults are in the elevator, the elevator is overloaded.