Question

rafael does a weekly exercise program consisting of cardiovascular work and weight training. each week, he exercises for at least 9 hours. he spends at most 6 hours on weight training. he spends at most 7 hours doing cardiovascular work. let x denote the time (in hours) that rafael spends doing cardiovascular work. let y denote the time (in hours) that he spends on weight training. shade the region corresponding to all values of x and y that satisfy these requirements.

Explanation:

Step1: Translate conditions to inequalities

He exercises for at least 9 hours, so $x + y\geq9$. He spends at most 6 hours on weight - training, so $y\leq6$. He spends at most 7 hours doing cardiovascular work, so $x\leq7$. Also, since time cannot be negative, $x\geq0$ and $y\geq0$.

Step2: Graph $x + y\geq9$

Rewrite it as $y\geq - x + 9$. The line $y=-x + 9$ has a y - intercept of 9 and a slope of - 1. Shade above this line.

Step3: Graph $y\leq6$

Draw a horizontal line at $y = 6$ and shade below it.

Step4: Graph $x\leq7$

Draw a vertical line at $x = 7$ and shade to the left of it.

Step5: Consider non - negativity

Shade the region in the first quadrant ($x\geq0,y\geq0$) that satisfies all the above inequalities.

Answer:

The shaded region is the intersection of the regions defined by $x + y\geq9$, $y\leq6$, $x\leq7$, $x\geq0$, and $y\geq0$. It is a polygon in the first - quadrant bounded by the intersection of the lines $x + y=9$, $y = 6$, $x = 7$, $x = 0$, and $y = 0$.