Question

Question was provided via image upload.

Explanation:

Identify the system of inequalities

We are given a system representing Colton's painting constraints:

• Time constraint: \(15x + 30y \ge 360\)
• Paint constraint: \(5x + 8y \le 100\)

Here, \(x\) represents the number of completed small figurines, and \(y\) represents the number of completed large figurines.

Analyze the boundary lines

Let's find the intercepts for each boundary line to understand the graph:

• For \(15x + 30y = 360\):
• If \(x = 0\), \(y = 12\).
• If \(y = 0\), \(x = 24\).
• Since the inequality is \(\ge\), the shaded region is above/right of this line (represented by the red region).

• For \(5x + 8y = 100\):
• If \(x = 0\), \(y = 12.5\).
• If \(y = 0\), \(x = 20\).
• Since the inequality is \(\le\), the shaded region is below/left of this line (represented by the blue region).

Determine the solution set

The solution to the system of inequalities is the intersection of the individual solutions.

• This is the overlapping region where both shaded areas meet.
• On the graph, this is the purple/overlapping region in the first quadrant where both conditions are simultaneously satisfied.
• Any point \((x, y)\) in this overlapping region represents a viable combination of small and large figurines Colton can paint.

Address the missing options

Since the multiple-choice options are cut off in the image, we will identify the mathematically correct statement that typically answers this standard problem:

• The solution set is represented by the overlapping shaded region (the purple region).
• Points in this region satisfy both the minimum time requirement of 360 minutes and the maximum paint limit of 100 milliliters.

Answer:

The correct option is the one that states:

• The overlapping shaded region (purple region) represents all possible combinations of small and large figurines Colton can paint to meet both his time and paint constraints. (Correct answer)