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Amanda owns a clothing store that sells graphic T-shirts. [tex]\( n \)[/tex] is the number of shirts she sells each month. The revenue function of her store is [tex]\( R = 15n \)[/tex]. The cost function of her store is [tex]\( C = 9n + 450 \)[/tex]. Using your calculator, what is the break-even point of Amanda's store?

A. [tex]\( n = 90 \)[/tex]
B. [tex]\( n = 75 \)[/tex]
C. [tex]\( n = 9 \)[/tex]
D. [tex]\( n = 15 \)[/tex]

Sagot :

GinnyAnsweridn
To determine the break-even point, we need to find the number of shirts [tex]\( n \)[/tex] where the total revenue equals the total cost. The revenue function and cost function are given as:

[tex]\[ r = 15n \][/tex]
[tex]\[ C = 9n + 450 \][/tex]

At the break-even point, revenue equals cost. Therefore, we set the two functions equal to each other:

[tex]\[ 15n = 9n + 450 \][/tex]

To solve for [tex]\( n \)[/tex], follow these steps:

1. Subtract [tex]\( 9n \)[/tex] from both sides of the equation to isolate the terms involving [tex]\( n \)[/tex] on one side:

[tex]\[ 15n - 9n = 450 \][/tex]

2. Simplify the left-hand side:

[tex]\[ 6n = 450 \][/tex]

3. Divide both sides of the equation by 6 to solve for [tex]\( n \)[/tex]:

[tex]\[ n = \frac{450}{6} \][/tex]

4. Calculate the value:

[tex]\[ n = 75 \][/tex]

Thus, the break-even point is when Amanda sells 75 T-shirts. Therefore, the correct answer is:

B. [tex]\( n = 75 \)[/tex]
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