Question

question 24 of 25
zane is mixing fruit punch in a 36 - cup punch bowl for a party. he plans to add at least 16 cups of fruit juice to the bowl before adding ginger ale.
identify the graph that represents the amounts of fruit juice and ginger ale he can use, and then identify a valid solution. (the graph shows the overlapping solution region only, not the shading for each inequality.)

Explanation:

Step1: Define Variables

Let \( x \) be the amount of fruit juice (in cups) and \( y \) be the amount of ginger ale (in cups).

Step2: Set Up Inequalities

1. The total capacity of the punch bowl is 36 cups, so \( x + y \leq 36 \).
2. He plans to add at least 16 cups of fruit juice, so \( x \geq 16 \).
3. Also, \( x \geq 0 \) and \( y \geq 0 \) (since amounts can't be negative).

Step3: Analyze the Graph

The solution region should be where \( x \geq 16 \), \( x + y \leq 36 \), \( x \geq 0 \), and \( y \geq 0 \) overlap. A valid solution could be when \( x = 20 \) (satisfying \( x \geq 16 \)), then from \( x + y \leq 36 \), we get \( y \leq 36 - 20 = 16 \). Let's take \( y = 10 \). So \( (20, 10) \) is a valid solution as \( 20 + 10 = 30 \leq 36 \) and \( 20 \geq 16 \).

Answer:

A valid solution is (20, 10) where \( x = 20 \) cups of fruit juice and \( y = 10 \) cups of ginger ale (the graph should show the region with \( x \geq 16 \), \( x + y \leq 36 \), \( x \geq 0 \), \( y \geq 0 \)).