Question

many elevators have a capacity of 4000 pounds. complete parts (a) through (c) below. number of adults cause the elevator to be overloaded. the inequality is 40x + 160y > 4000. 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.

Explanation:

Step1: Rewrite the inequality in slope - intercept form.

First, start with $40x + 160y>4000$. Subtract $40x$ from both sides: $160y>- 40x + 4000$. Then divide each term by 160: $y>-\frac{40}{160}x+\frac{4000}{160}$, which simplifies to $y >-\frac{1}{4}x + 25$.

Step2: Find the boundary line.

The boundary line of the inequality $y>-\frac{1}{4}x + 25$ is the equation $y =-\frac{1}{4}x + 25$. The $y$-intercept is $b = 25$ (the point $(0,25)$) and the slope $m=-\frac{1}{4}$.

Step3: Determine which side of the boundary line to shade.

Since the inequality is $y>-\frac{1}{4}x + 25$, we can test a point not on the line. A common test - point is $(0,0)$. Substitute $x = 0$ and $y = 0$ into the inequality: $0>-\frac{1}{4}(0)+25$, or $0>25$, which is false. So, we shade the side of the line that does not contain the origin. Also, since $x\geq0$ and $y\geq0$, we limit the graph to the first quadrant.

To graph:

1. Plot the $y$-intercept at the point $(0,25)$.
2. Use the slope $-\frac{1}{4}$ to find another point. From $(0,25)$, move 4 units to the right and 1 unit down to get the point $(4,24)$.
3. Draw a dashed line (because the inequality is $>$ and not $\geq$) through the points $(0,25)$ and $(4,24)$.
4. Shade the region in the first quadrant above the line.

Answer:

Graph a dashed line with $y$-intercept 25 and slope $-\frac{1}{4}$ in the first quadrant and shade the region above the line.