To find the number of units [tex]\( x \)[/tex] that must be sold to break even, we need to set the cost function [tex]\( C(x) \)[/tex] equal to the revenue function [tex]\( R(x) \)[/tex] and solve for [tex]\( x \)[/tex].
The cost function is given by:
[tex]\[ C(x) = 73x + 106,000 \][/tex]
The revenue function is given by:
[tex]\[ R(x) = 173x \][/tex]
To break even, the cost must be equal to the revenue:
[tex]\[ C(x) = R(x) \][/tex]
So:
[tex]\[ 73x + 106,000 = 173x \][/tex]
Now, we need to solve this equation for [tex]\( x \)[/tex].
First, we isolate [tex]\( x \)[/tex] on one side of the equation. We do this by subtracting [tex]\( 73x \)[/tex] from both sides:
[tex]\[ 106,000 = 173x - 73x \][/tex]
Simplify the right side:
[tex]\[ 106,000 = 100x \][/tex]
Next, solve for [tex]\( x \)[/tex] by dividing both sides of the equation by 100:
[tex]\[ x = \frac{106,000}{100} \][/tex]
[tex]\[ x = 1060 \][/tex]
Therefore, to break even, [tex]\( 1060 \)[/tex] units must be sold.
