To solve this problem, we need to plot the cost function [tex]\( C = 6n + 200 \)[/tex] and the revenue function [tex]\( r = 14n \)[/tex] and identify the break-even point, where the cost equals the revenue.
### Step-by-Step Solution
1. Understand the Functions:
- The cost function is [tex]\( C(n) = 6n + 200 \)[/tex].
- The revenue function is [tex]\( r(n) = 14n \)[/tex].
2. Identify the Break-Even Point:
- The break-even point is where the cost function equals the revenue function:
[tex]\[
6n + 200 = 14n
\][/tex]
3. Solve for [tex]\( n \)[/tex]:
1. Subtract [tex]\( 6n \)[/tex] from both sides:
[tex]\[
200 = 8n
\][/tex]
2. Divide both sides by 8:
[tex]\[
n = \frac{200}{8} = 25
\][/tex]
So, the break-even point is [tex]\( n = 25 \)[/tex].
4. Plot the Functions:
- Create a graph with the x-axis representing the number of buckets of corn sold ([tex]\( n \)[/tex]), and the y-axis representing the cost or revenue.
- Plot the cost function ([tex]\( C = 6n + 200 \)[/tex]) and the revenue function ([tex]\( r = 14n \)[/tex]).
5. Checking Graphically:
- To plot these functions, choose a range of [tex]\( n \)[/tex] (say from 0 to 50) and calculate corresponding [tex]\( C \)[/tex] and [tex]\( r \)[/tex].
For example:
- For [tex]\( n = 0 \)[/tex], [tex]\( C = 200 \)[/tex] and [tex]\( r = 0 \)[/tex].
- For [tex]\( n = 10 \)[/tex], [tex]\( C = 260 \)[/tex] and [tex]\( r = 140 \)[/tex].
- For [tex]\( n = 25 \)[/tex], [tex]\( C = 350 \)[/tex] and [tex]\( r = 350 \)[/tex].
- For [tex]\( n = 50 \)[/tex], [tex]\( C = 500 \)[/tex] and [tex]\( r = 700 \)[/tex].
When you plot these points, the graph should intersect at [tex]\( n = 25 \)[/tex].
### Conclusion
On the graph, the cost function [tex]\( C = 6n + 200 \)[/tex] and the revenue function [tex]\( r = 14n \)[/tex] will intersect at the break-even point. The calculations show that the break-even point is at [tex]\( n = 25 \)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{n = 25} \][/tex]
This corresponds to option C.
