Let's solve this step-by-step.
1. Define the variables: Let [tex]\( x \)[/tex] be the number of sculptures Rob makes and sells. Let [tex]\( y \)[/tex] be the dollars involved, either cost or revenue.
2. Define the cost equation:
Rob's initial cost is [tex]\( \$200 \)[/tex].
It costs Rob [tex]\( \$10 \)[/tex] for materials to make each sculpture.
Therefore, the total cost [tex]\( y \)[/tex] for making [tex]\( x \)[/tex] sculptures is given by:
[tex]\[
y = 200 + 10x
\][/tex]
3. Define the revenue equation:
Rob sells each sculpture for [tex]\( \$50 \)[/tex].
Therefore, the total revenue [tex]\( y \)[/tex] from selling [tex]\( x \)[/tex] sculptures is given by:
[tex]\[
y = 50x
\][/tex]
4. Set up the equations to find the break-even point:
At the break-even point, costs equal revenue. Thus, we set the cost equation equal to the revenue equation:
[tex]\[
200 + 10x = 50x
\][/tex]
5. Solve for [tex]\( x \)[/tex]:
Isolate [tex]\( x \)[/tex] by moving all terms involving [tex]\( x \)[/tex] to one side of the equation:
[tex]\[
200 + 10x = 50x
\][/tex]
Subtract [tex]\( 10x \)[/tex] from both sides:
[tex]\[
200 = 40x
\][/tex]
Divide both sides by 40 to solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{200}{40} = 5
\][/tex]
Therefore, Rob must sell [tex]\( \boxed{5} \)[/tex] sculptures to break even.
