To determine the number of guitars the company needs to sell to start making a profit, we need to calculate the break-even point. The break-even point is where the total revenue [tex]\(R(x)\)[/tex] equals the total cost [tex]\(C(x)\)[/tex].
Given the revenue function:
[tex]\[ R(x) = 120x \][/tex]
And the cost function:
[tex]\[ C(x) = 100x + 1,840 \][/tex]
To find the break-even point, we set the revenue equal to the cost:
[tex]\[ 120x = 100x + 1,840 \][/tex]
Now, solve for [tex]\(x\)[/tex]:
1. Subtract [tex]\(100x\)[/tex] from both sides:
[tex]\[ 120x - 100x = 1,840 \][/tex]
2. Simplify the equation:
[tex]\[ 20x = 1,840 \][/tex]
3. Divide both sides by 20:
[tex]\[ x = \frac{1,840}{20} \][/tex]
[tex]\[ x = 92 \][/tex]
At the break-even point, the company needs to sell at least 92 guitars.
To find the total revenue at the break-even point:
1. Substitute [tex]\(x = 92\)[/tex] into the revenue function [tex]\(R(x)\)[/tex]:
[tex]\[ R(92) = 120 \cdot 92 \][/tex]
[tex]\[ R(92) = 11,040 \][/tex]
Therefore, the company needs to sell at least [tex]\( \boxed{92} \)[/tex] guitars for a total revenue of [tex]\( \$ \boxed{11,040} \)[/tex] to start making a profit.
