$$ 1

Question

$$ 1<\quad $$ Points: 0 of 1 The manager of a small company that produces roof tile has determined that the total cost in dollars, $$ \mathrm{C}(\mathrm{x}) $$, of producing $$ x $$ units of tile is given by $$ \mathrm{C}(\mathrm{x})=200 \mathrm{x}+900 $$, while the revenue in dollars, $$ R(x) $$, from the sale of $$ x $$ units of tile is given by $$ \mathrm{R}(\mathrm{x})=230 \mathrm{x} $$. Find the break-even point and the cost and revenue at the break-even point.

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Break-even point: $$ x = 30 $$ units, Cost: $$ 6900 $$ dollars, Revenue: $$ 6900 $$ dollars. To find the break-even point, we need to determine the value of $$ x $$ where the total cost $$ C(x) $$ equals the revenue $$ R(x) $$. The equations given are: $$ C(x) = 200x + 900 $$ $$ R(x) = 230x $$ Setting the cost equal to the revenue: $$ 200x + 900 = 230x $$ Now, we can solve for $$ x $$: 1. Subtract $$ 200x $$ from both sides: $$ 900 = 230x - 200x $$ 2. Simplify: $$ 900 = 30x $$ 3. Divide both sides by 30: $$ x = \frac{900}{30} = 30 $$ So, the break-even point is at $$ x = 30 $$ units. Next, we need to find the cost and revenue at the break-even point: 1. Calculate the cost at $$ x = 30 $$: $$ C(30) = 200(30) + 900 = 6000 + 900 = 6900 $$ 2. Calculate the revenue at $$ x = 30 $$: $$ R(30) = 230(30) = 6900 $$ Thus, at the break-even point: - The break-even point is $$ x = 30 $$ units. - The cost at the break-even point is $$ C(30) = 6900 $$ dollars. - The revenue at the break-even point is $$ R(30) = 6900 $$ dollars. In summary: - Break-even point: $$ x = 30 $$ - Cost at break-even: $$ C(30) = 6900 $$ - Revenue at break-even: $$ R(30) = 6900 $$

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