Question

Produced and sold each day.

R(x)=50x
C(x)=32.5x+52,500
​
(a) Find the firm's 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 earn a profit.

Answer 1

(a) Break-even point: x = 3000

(b) x > 3000

Explanation:

(a)

To find the break-even point, we set R(x) equal to C(x) and solve for x:

R(x) = C(x)

(50x = 32.5x + 52500) - 32.5

(17.5x = 52500) / 17.5

x = 3000

Thus, the firm must sell 3000 units to break even.

(b)

• Since the break-even point is the point at which revenue--R(x) equals cost--C(x), there is no profit made (no gains) and no money loss (cost).

• In order to make a profit, a business must sell more than the break-even point.

Thus, the business must sell more than 3000 units to make a profit so x > 3000 is the answer.

• We can see this by plugging in any integer greater than 3000 for x in R(x) and C(x)--it must be an integer since the business can't make part of an item--and seeing if R(x) is greater than C(x).

Let's try 3001 for x:

R(3001) > C(3001)

50(3001) > 52.5(3001) + 52500

150050 > 97532.5 + 52500

150050 > 150032.5