The cost function of a corn stand is [tex]C = 6n + 200[/tex], where [tex]n[/tex] is the number of buckets of corn sold. The revenue function of this corn stand is [tex]R = 14n[/tex].

Create a graph to show each of these functions. According to your graph, what is the break-even point for this corn stand?

A. [tex]n = 14[/tex]
B. [tex]n = 28[/tex]
C. [tex]n = 25[/tex]
D. [tex]n = 6[/tex]



Answer :

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.