operties of Linear Question 22, 2.1.49 HW Score: $$ 26.75 \%, 6.69 $$ of 25 points ppilications Pat2 of 2 Points: 0 of 1 Save 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. $$ \begin{array}{l} R(x)=25 x \ C(x)=12.5 x+50,000 \end{array} $$ (a) Find the fimis break even point; that is, find $$ x $$ so that $$ R=C $$. (b) Find the values of $$ x $$ such that $$ R(x)C(x) $$. This represents the number of units that the company must sell to eam a profit. (a) $$ x=4000 $$ (Type a whole number.) (b) Solve the inequality for $$ x $$. Type the correct inequality symbol in the first answer box below, and type an integer in the second answer box. $$ x $$ $$ \square $$

Question

operties of Linear Question 22, 2.1.49 HW Score: $$ 26.75 \%, 6.69 $$ of 25 points ppilications Pat2 of 2 Points: 0 of 1 Save 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. $$ \begin{array}{l} R(x)=25 x \ C(x)=12.5 x+50,000 \end{array} $$ (a) Find the fimis break even point; that is, find $$ x $$ so that $$ R=C $$. (b) Find the values of $$ x $$ such that $$ R(x)>C(x) $$. This represents the number of units that the company must sell to eam a profit. (a) $$ x=4000 $$ (Type a whole number.) (b) Solve the inequality for $$ x $$. Type the correct inequality symbol in the first answer box below, and type an integer in the second answer box. $$ x $$ $$ \square $$

UpStudy AI · Accepted Answer

To solve the problem, we need to follow the steps outlined in the prompt.

### Given:
- Revenue function: $$ R(x) = 25x $$
- Cost function: $$ C(x) = 12.5x + 50,000 $$

### (a) Find the break-even point where $$ R = C $$.

1. Set the revenue equal to the cost:
   $$
   25x = 12.5x + 50,000
   $$

2. Rearrange the equation to isolate $$ x $$:
   $$
   25x - 12.5x = 50,000
   $$
   $$
   12.5x = 50,000
   $$

3. Solve for $$ x $$:
   $$
   x = \frac{50,000}{12.5}
   $$

Now, let's calculate $$ x $$.

### (b) Find the values of $$ x $$ such that $$ R(x) > C(x) $$.

1. Set up the inequality:
   $$
   25x > 12.5x + 50,000
   $$

2. Rearrange the inequality:
   $$
   25x - 12.5x > 50,000
   $$
   $$
   12.5x > 50,000
   $$

3. Solve for $$ x $$:
   $$
   x > \frac{50,000}{12.5}
   $$

Now, let's perform the calculations for both parts.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{50000}{12.5}$$
- step1: Convert the expressions:
  $$\frac{50000}{\frac{25}{2}}$$
- step2: Multiply by the reciprocal:
  $$50000\times \frac{2}{25}$$
- step3: Reduce the numbers:
  $$2000\times 2$$
- step4: Multiply the numbers:
  $$4000$$
Calculate or simplify the expression $$ 50000/12.5 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{50000}{12.5}$$
- step1: Convert the expressions:
  $$\frac{50000}{\frac{25}{2}}$$
- step2: Multiply by the reciprocal:
  $$50000\times \frac{2}{25}$$
- step3: Reduce the numbers:
  $$2000\times 2$$
- step4: Multiply the numbers:
  $$4000$$
### Results:

(a) The break-even point is:
$$
x = 4000
$$

(b) The inequality for profit is:
$$
x > 4000
$$

### Final Answers:
- (a) $$ x = 4000 $$
- (b) $$ x $$ $$ > $$ $$ 4000 $$

This means the company must sell more than 4000 units to earn a profit.
(a) The break-even point is 4000 units. (b) The company must sell more than 4000 units to earn a profit.

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