Question

a surfboard - shaper has to limit the cost of development and production to $219 per surfboard. he has already spent $3,843 on equipment for the boards. the development and production costs are $138 per board. the cost per board is $\frac{138x + 3843}{x}$ dollars. determine the number of boards that must be sold to limit the final cost per board to $219.
how many boards must be sold to limit the cost per board to $219?

Explanation:

Step1: Set up the cost - per - board equation

Let $x$ be the number of boards. The total cost is the sum of the fixed cost ($31443$) and the variable cost ($138x$). The cost per board is $\frac{138x + 31443}{x}$. We want this cost per board to be $259$. So, we set up the equation $\frac{138x+31443}{x}=259$.

Step2: Multiply both sides by $x$

Multiply both sides of the equation $\frac{138x + 31443}{x}=259$ by $x$ to get $138x+31443 = 259x$.

Step3: Rearrange the equation

Subtract $138x$ from both sides: $31443=259x - 138x$.

Step4: Simplify the right - hand side

$259x-138x = 121x$, so the equation becomes $31443 = 121x$.

Step5: Solve for $x$

Divide both sides by $121$: $x=\frac{31443}{121}=259.86$. Since we can't sell a fraction of a board, we need to sell $260$ boards.

Answer:

$260$