Question

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a firm will break even (no profit and no loss) as long as revenue just equals cost. the value of x (the number of items produced and sold) where c(x)=r(x) is called the break - even point. assume that the below table can be expressed as a linear function. find (a) the cost function, (b) the revenue function, and (c) the profit function. (d) find the break - even point and decide whether the product should be produced, given the restrictions on sales.

fixed cost	variable cost	price of item

according to the restriction, no more than 20 units can be sold.
(a) the cost function is c(x)=300 + 15x
(simplify your answer.)
(b) the revenue function is r(x)=25x
(simplify your answer.)
(c) the profit function is p(x)=10x - 300
(simplify your answer.)
(d) select the correct choice below and fill in the answer box within your choice.
(type a whole number.)
○ a. the break - even point is □ units. thus, the product should be produced, given the restriction on sales.
● b. the break - even point is □ units. thus, the product should not be produced, given the restriction on sales.

Explanation:

Step1: Set cost equal to revenue

To find the break - even point, we set \(C(x)=R(x)\). We know that \(C(x) = 300+15x\) and \(R(x)=25x\). So we set up the equation \(300 + 15x=25x\).

Step2: Solve for x

Subtract \(15x\) from both sides of the equation:
\(300+15x - 15x=25x-15x\)
\(300 = 10x\)
Then divide both sides by 10:
\(x=\frac{300}{10}=30\)

Step3: Compare with sales restriction

The sales restriction is that no more than 20 units can be sold. Since the break - even point \(x = 30\) is greater than 20, the product should not be produced.

Answer:

The break - even point is \(\boldsymbol{30}\) units. Thus, the product should not be produced, given the restriction on sales (Option B).