picculions Pant 10 Points 0 of 1 The point at which a company's profits equal zere is called the company's break even point Lef R represent a company's revenve, let Crepresent the company's costs, and let x represent the number of units produced and sold each day \[ \begin{array}{l} \mathrm{R}(x)=25 x \\ \mathrm{C}(x)=126 x+50.000 \end{array} \] (a) Find the firm's break even point, that is, find \( \times 50 \) that \( R=C \). (b) Find the values of \( x \) such that \( \mathrm{R}(x)>\mathrm{C}(x) \) This represents the number of units that the company must sell to eam a profit. (a) \( x= \) \( \square \) (Type a whole number)

To find the break-even point, we need to set revenue equal to costs: \[ R(x) = C(x) \] Substituting the given functions: \[ 25x = 126x + 50,000 \] Now, we can solve for \(x\): \[ 25x - 126x = 50,000 \\ -101x = 50,000 \\ x = \frac{-50,000}{-101} \\ x \approx 495.05 \] Since \(x\) must be a whole number, round it up to get \(x = 496\). Now for part (b), we need to find \(x\) such that: \[ R(x) > C(x) \] This can be expressed as: \[ 25x > 126x + 50,000 \] Simplifying this gives: \[ -101x > 50,000 \\ x < -\frac{50,000}{101} \\ x < -495.05 \] Since negative production isn’t feasible, the company must sell more than 496 units to make a profit. So, the answers are: (a) \( x = 496 \) (b) The company must sell more than 496 units to earn a profit.