To determine the inequality that describes the relationship between the number of fruit-filled cakes (x) and the number of plain cakes (y), we can start by setting up the equation based on the sales requirements.
The cost of a fruit-filled cake is $120 and the cost of a plain cake is $90. To meet the sales goal of at least $3,600, we can write the inequality:
\[ 120x + 90y \geq 3600 \]
To find the boundary line of the inequality, we can set it up as an equation:
\[ 120x + 90y = 3600 \]
Next, we can rearrange this equation to clearly see the intercepts and derive the points for graphing:
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Finding the y-intercept (when \(x = 0\)): \[ 90y = 3600 \quad \Rightarrow \quad y = \frac{3600}{90} = 40 \] So, the y-intercept is at (0, 40).
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Finding the x-intercept (when \(y = 0\)): \[ 120x = 3600 \quad \Rightarrow \quad x = \frac{3600}{120} = 30 \] So, the x-intercept is at (30, 0).
The line that we get from \(120x + 90y = 3600\) passes through the points (0, 40) and (30, 0).
Now, because the original constraint was \(120x + 90y \geq 3600\), we need to shade the region where the sales are greater than or equal to $3,600. This means we need to shade above the line corresponding to our boundary.
Next, we consider whether our line is solid or dashed. Since the inequality includes "greater than or equal to," we will use a solid line to represent the points on the line as included in the solution set.
Given these considerations, the correct answer is:
A. The graph shows 'Fruit-filled cakes' from 0 to 50 along the x-axis and 'Plain cakes' from 0 to 50 along the y-axis. A solid line passes through the points (0, 40) and (30, 0). The area to the right of the line is shaded.