To represent Samantha's fundraising situation graphically, we need to formulate the relationship based on the information provided regarding corporate and individual sponsorships.
Let's define:
- \( c \): the number of corporate sponsorships.
- \( i \): the number of individual sponsorships.
Each corporate sponsorship raises $100.00, so the total amount raised from corporate sponsorships can be expressed as \( 100c \).
Each individual sponsorship raises $25.00, thus the total amount from individual sponsorships is \( 25i \).
Samantha needs to raise at least $500, so we can set up the following inequality:
\[ 100c + 25i \geq 500 \]
We can simplify this inequality by dividing all terms by 25:
\[ 4c + i \geq 20 \]
This inequality represents a half-plane in a coordinate system where \( c \) (corporate sponsorships) is on the x-axis and \( i \) (individual sponsorships) is on the y-axis.
Steps to graph the inequality:
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First, identify the equality line \( 4c + i = 20 \):
- When \( c = 0 \): \( 4(0) + i = 20 \) → \( i = 20 \) (point (0, 20))
- When \( i = 0 \): \( 4c + 0 = 20 \) → \( c = 5 \) (point (5, 0))
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Plot the line connecting points \( (0, 20) \) and \( (5, 0) \).
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Since the inequality is \( \geq \), you will shade above this line to represent all combinations of corporate and individual sponsorships that satisfy the requirement of raising at least $500.
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Identify that \( c \) and \( i \) must be non-negative integers, so you'll be working in the first quadrant where both \( c \) and \( i \) are 0 or greater.
Resulting Graph:
The graph you are looking for would contain:
- A line from (0, 20) to (5, 0) representing the boundary \( 4c + i = 20 \).
- Shading above this line (towards the top right) showing the feasible solutions where the total amount raised is at least $500.
- The axes indicating only non-negative values for \( c \) and \( i \).
If you provide any options of graphs, I can help you identify which one corresponds to this description.