Question

daniel is printing out copies of a presentation. it takes 3 minutes to print a color copy and 1 minute to print a black-and-white copy. he needs to print at least 6 copies and must have the copies completed in no more than 12 minutes. if the solution region represents the number of color copies and black-and-white copies that daniel can print, determine which graph represents the solution set to the system of inequalities representing this situation. a. graph a: number of black-and-white copies (y-axis), number of color copies (x-axis), with a shaded region b. graph b: number of black-and-white copies (y-axis), number of color copies (x-axis), with a shaded region

Explanation:

Step1: Define Variables

Let \( x \) = number of color copies, \( y \) = number of black - and - white copies.

Step2: Time Constraint Inequality

Time to print color copies: \( 3x \) minutes, time to print black - and - white copies: \( y \) minutes. Total time \( \leq12 \), so \( 3x + y\leq12 \), or \( y\leq - 3x + 12 \).

Step3: Minimum Copies Constraint

Total copies \( x + y\geq6 \), or \( y\geq - x+6 \). Also, \( x\geq0,y\geq0 \) (non - negative number of copies).

Step4: Analyze Graph Axes

In graph A, x - axis is number of color copies, y - axis is number of black - and - white copies. In graph B, x - axis is number of black - and - white copies, y - axis is number of color copies (incorrect axis labeling for the problem's context as per the time and copy number relationships).

Step5: Check Constraints on Graph A

For \( 3x + y\leq12 \), when \( x = 0 \), \( y = 12 \); when \( y = 0 \), \( x = 4 \). For \( x + y\geq6 \), when \( x = 0 \), \( y = 6 \); when \( y = 0 \), \( x = 6 \). The shaded region in graph A should satisfy these inequalities. The lines and shaded region in graph A align with the inequalities \( 3x + y\leq12 \) and \( x + y\geq6 \) (and non - negativity), while graph B has incorrect axis mapping (swapped x and y for the key constraints).

Answer:

A. The graph labeled A