To find the number of units [tex]\( x \)[/tex] that must be produced and sold to break even, we need to determine where the cost function [tex]\( C(x) \)[/tex] equals the revenue function [tex]\( R(x) \)[/tex].
The cost function is given by:
[tex]\[ C(x) = 15x + 36,000 \][/tex]
The revenue function is given by:
[tex]\[ R(x) = 18x \][/tex]
To break even, the cost and revenue must be equal:
[tex]\[ C(x) = R(x) \][/tex]
So, we set the equations equal to each other and solve for [tex]\( x \)[/tex]:
[tex]\[ 15x + 36,000 = 18x \][/tex]
First, subtract [tex]\( 15x \)[/tex] from both sides to isolate the term involving [tex]\( x \)[/tex] on one side of the equation:
[tex]\[ 36,000 = 18x - 15x \][/tex]
Simplify the right-hand side:
[tex]\[ 36,000 = 3x \][/tex]
Next, solve for [tex]\( x \)[/tex] by dividing both sides of the equation by 3:
[tex]\[ x = \frac{36,000}{3} \][/tex]
[tex]\[ x = 12,000 \][/tex]
Therefore, the number of units that must be produced and sold to break even is:
[tex]\[ \boxed{12,000} \][/tex] units.
