Samantha is raising money for a local charity. She is required to raise at least $500.00 through corporate and individual sponsorships. Each corporate sponsorship raises $100.00, and each individual sponsorship raises $25.00.

Question

Samantha is raising money for a local charity. She is required to raise at least $500.00 through corporate and individual sponsorships. Each corporate sponsorship raises $100.00, and each individual sponsorship raises $25.00.
Which graph represents the situation?

GPT-4o mini · Answer

To represent Samantha’s fundraising situation with a graph, we can use a coordinate plane where one axis represents the number of corporate sponsorships (let's say this is the x-axis) and the other axis represents the number of individual sponsorships (let's say this is the y-axis).

Step 1: Set Up the Equation

The total amount raised from corporate and individual sponsorships can be expressed with the following equation:

\[ 100x + 25y \geq 500 \]

Where:

$ x $ = Number of corporate sponsorships
$ y $ = Number of individual sponsorships

Step 2: Simplify the Equation

We can simplify this inequality. First, we can divide the entire inequality by 25:

\[ 4x + y \geq 20 \]

Step 3: Identify the Line

The line that represents the equation $ 4x + y = 20 $ can be determined by finding intercepts:

If $ x = 0 $, then $ y = 20 $ (y-intercept).
If $ y = 0 $, then $ 4x = 20 $, which gives $ x = 5 $ (x-intercept).

Step 4: Graph the Line

You can plot the points $(0, 20)$ and $(5, 0)$ on a graph, and then draw the line passing through these points. Since the inequality is $ \geq $, you will shade the area above (including the line) which represents all the combinations of corporate and individual sponsorships that meet or exceed the target of $500.

Step 5: Axes Limits

Since both $ x $ and $ y $ must be non-negative, you will restrict your graph to the first quadrant.

Final Graph

You will look for a graph that:

Shows the line $ 4x + y = 20 $.
Shades the region above this line, including the line itself.
Contains only points in the first quadrant where both $ x $ and $ y $ are non-negative.

This is the graph that represents Samantha's fundraising situation.

Answer

is the line dashed or not

GPT-4o mini · Answer

In this case, the line should be solid, not dashed.

Since the inequality is $4x + y \geq 20$, the solid line indicates that points on the line itself (where $4x + y = 20$) are included in the solution set. This means combinations of corporate and individual sponsorships that yield exactly $500 are valid, along with those combinations that yield more than $500, which is represented by the shaded region above the line.

So, to summarize:

The line $4x + y = 20$ should be solid.
The area above the line (where $4x + y > 20$) should be shaded.