ties of Linear ations The point at which a company's profits equal zero is called the company's break even point. Let R represent a company's revenue, let C represent the company's costs, and let x represent the number of units produced and sold each day. \( \mathrm{R}(\mathrm{x})=25 \mathrm{x} \). \( 26.75 \%, 6.69 \) of 25 points \( \mathrm{C}(\mathrm{x})=12.5 \mathrm{x}+50,000 \) (a) Find the firm's break-even point, that is, find x so that \( \mathrm{R}=\mathrm{C} \). (b) Find the values of x such that \( \mathrm{R}(\mathrm{x})>\mathrm{C}(\mathrm{x}) \). This represents the number of units that the company must sell to earn a profit. (a) \( \mathrm{x}=\square \) (Type a whole number.)

To find the firm's break-even point, we need to set the revenue equal to the costs: \[ R(x) = C(x) \] Using the provided functions: \[ 25x = 12.5x + 50,000 \] To solve for x, we can subtract 12.5x from both sides: \[ 25x - 12.5x = 50,000 \] This gives us: \[ 12.5x = 50,000 \] Now, dividing both sides by 12.5 results in: \[ x = \frac{50,000}{12.5} = 4000 \] So, the firm's break-even point is \( x = 4000 \). For part (b), we need to find the values of \( x \) such that \( R(x) > C(x) \): \[ 25x > 12.5x + 50,000 \] Subtracting \( 12.5x \) from both sides gives: \[ 12.5x > 50,000 \] Now, dividing by 12.5 results in: \[ x > 4000 \] Thus, the values of \( x \) for which the revenue is greater than the costs are \( x > 4000 \). (a) \( \mathrm{x}=4000 \)