Question

question 7 of 10
misha wants to spend some time playing guitar and using the computer before dinner.

dinner will be ready in 45 minutes or less.
he wants to spend at least 15 minutes playing guitar.

which graph represents the amounts of time misha can spend playing guitar and using the computer?

Explanation:

To solve this, we define variables: let \( g \) be guitar time (minutes) and \( c \) be computer time (minutes).

Step 1: Total Time Constraint

Dinner is ready in 45 minutes or less, so \( g + c \leq 45 \). This is a line \( g + c = 45 \) (slope - 1, intercepts (45,0) and (0,45)) with the region below the line shaded (since \( g,c \geq 0 \) for time).

Step 2: Guitar Time Constraint

He wants at least 15 minutes on guitar, so \( g \geq 15 \). This is a vertical line \( g = 15 \) with the region to the right of the line shaded (since \( g \) must be 15 or more).

The correct graph will have:

• A solid line for \( g + c = 45 \) (since the inequality is \( \leq \)) with shading below it.
• A solid line for \( g = 15 \) (since the inequality is \( \geq \)) with shading to the right of it.
• The feasible region is where both shadings overlap, with \( g \geq 15 \), \( c \geq 0 \), and \( g + c \leq 45 \).

To identify the graph, look for these two boundary lines (solid, as inequalities are non - strict) and the correct shaded regions. The feasible region will be a polygon bounded by \( g = 15 \), \( c = 0 \), and \( g + c = 45 \).

(Note: Since the actual graphs are not shown here, the key is to check for the two constraints: \( g\geq15 \) (vertical line at \( g = 15 \), shaded right) and \( g + c\leq45 \) (line with negative slope, shaded below), with \( c\geq0 \).)

If we assume the options have these features, the graph with:

• Vertical solid line at \( g = 15 \), shaded right.
• Solid line \( g + c = 45 \), shaded below.
• Overlapping shaded region (e.g., a triangle with vertices at (15,0), (15,30), and (45,0)) is the correct one.

(In a typical multiple - choice setup, the correct graph will match these two constraints’ visual representations.)

Answer:

To solve this, we define variables: let \( g \) be guitar time (minutes) and \( c \) be computer time (minutes).

Step 1: Total Time Constraint

Dinner is ready in 45 minutes or less, so \( g + c \leq 45 \). This is a line \( g + c = 45 \) (slope - 1, intercepts (45,0) and (0,45)) with the region below the line shaded (since \( g,c \geq 0 \) for time).

Step 2: Guitar Time Constraint

He wants at least 15 minutes on guitar, so \( g \geq 15 \). This is a vertical line \( g = 15 \) with the region to the right of the line shaded (since \( g \) must be 15 or more).

The correct graph will have:

• A solid line for \( g + c = 45 \) (since the inequality is \( \leq \)) with shading below it.
• A solid line for \( g = 15 \) (since the inequality is \( \geq \)) with shading to the right of it.
• The feasible region is where both shadings overlap, with \( g \geq 15 \), \( c \geq 0 \), and \( g + c \leq 45 \).

To identify the graph, look for these two boundary lines (solid, as inequalities are non - strict) and the correct shaded regions. The feasible region will be a polygon bounded by \( g = 15 \), \( c = 0 \), and \( g + c = 45 \).

(Note: Since the actual graphs are not shown here, the key is to check for the two constraints: \( g\geq15 \) (vertical line at \( g = 15 \), shaded right) and \( g + c\leq45 \) (line with negative slope, shaded below), with \( c\geq0 \).)

If we assume the options have these features, the graph with:

• Vertical solid line at \( g = 15 \), shaded right.
• Solid line \( g + c = 45 \), shaded below.
• Overlapping shaded region (e.g., a triangle with vertices at (15,0), (15,30), and (45,0)) is the correct one.

(In a typical multiple - choice setup, the correct graph will match these two constraints’ visual representations.)