Problem 81 - College Algebra

Question

Many elevators have a capacity of 2000 pounds. a. If a child averages 50 pounds and an adult 150 pounds, write an inequality that describes when \(x\) children and \(y\) adults will cause the elevator to be overloaded. 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 derived inequality equation is \(50x + 150y > 2000\). The shaded area in the graph represents position of children and adults that will cause the elevator to overload. The ordered pair (20,10) satisfies the inequality indicating that 20 children and 10 adults will cause the elevator to overload.

Step 1: Formulate the inequality

Given that a child averages 50 pounds and an adult 150 pounds, the total weight for \(x\) children and \(y\) adults can be represented as \(50x + 150y\). The elevator is overloaded when this weight exceeds 2000 pounds. So, the required inequality is \(50x + 150y > 2000\).

Step 2: Graph the inequality

To graph the inequality, it would be simpler to first graph the equation equivalent to it, which is \(50x + 150y = 2000\). To do this, find the x- and y- intercepts. Setting \(y=0\), solve the equivalent equation to find the x-intercept, which gives \(x = 2000 / 50 = 40\), so the x-intercept is (40,0). Similarly set \(x = 0\) and solve for y to find the y-intercept: \(y = 2000 / 150\), which is approximately 13.33, giving the y-intercept (0,13.33). Plot these two points on the graph and draw a straight line through them. Since the inequality is 'greater than', the portion above the line is shaded, indicating the area representing the overloaded capacity. This means that every point in the shaded area represents the number of children and adults that will overload the elevator.

Step 3: Select an ordered pair

Any pair in the shaded area will satisfy the inequality. For instance, using the pair (20,10), interpreting this in the context of the problem means, if there are 20 children and 10 adults in the elevator, it will be overloaded. This works as you can verify: \(50(20) + 150(10) = 2000\), which is the limit of the elevator's capacity, any additional weight would result in overloading.