Let's analyze the given cost and revenue functions step-by-step to find the break-even point for the peach stand.
### Step-by-Step Solution:
#### 1. Define the Cost Function
The cost function [tex]\( C \)[/tex] for the peach stand is given by:
[tex]\[ C = 9n + 165 \][/tex]
where [tex]\( n \)[/tex] is the number of buckets of peaches sold.
#### 2. Define the Revenue Function
The revenue function [tex]\( r \)[/tex] for the peach stand is given by:
[tex]\[ r = 20n \][/tex]
#### 3. Graph the Functions
To find the break-even point, we need to graph both functions [tex]\( C \)[/tex] and [tex]\( r \)[/tex] and determine where they intersect.
- The cost function [tex]\( C = 9n + 165 \)[/tex] is a linear function with a y-intercept at 165 and a slope of 9. This means the graph of the cost function increases steeply with [tex]\( n \)[/tex] starting from 165.
- The revenue function [tex]\( r = 20n \)[/tex] is also a linear function with no y-intercept (i.e., it goes through the origin) and a slope of 20. This means that the graph of the revenue function increases faster than the cost function as [tex]\( n \)[/tex] increases.
#### 4. Determine the Break-Even Point
The break-even point is where the cost equals the revenue. In other words, we need to find [tex]\( n \)[/tex] where:
[tex]\[ 9n + 165 = 20n \][/tex]
Rearranging the equation to solve for [tex]\( n \)[/tex]:
[tex]\[ 165 = 20n - 9n \][/tex]
[tex]\[ 165 = 11n \][/tex]
[tex]\[ n = \frac{165}{11} \][/tex]
[tex]\[ n = 15 \][/tex]
#### 5. Conclusion
The break-even point, where the cost of producing buckets of peaches equals the revenue from selling them, is at [tex]\( n = 15 \)[/tex]. Therefore, the correct option is:
[tex]\[ \boxed{n = 15} \][/tex]
Thus, according to the graph, the break-even point for this peach stand is when [tex]\( n = 15 \)[/tex] buckets of peaches are sold. Thus, the correct answer is:
C. [tex]\( n = 15 \)[/tex]
