Question

The total revenue function for a product is given by

R=655x

​dollars, and the total cost function for this same product is given by

C=19,250+70x+x2​,

where C is measured in dollars. For both​ functions, the input x is the number of units produced and sold.

a. Form the profit function for this product from the two given functions.

b. What is the profit when

25

units are produced and​ sold?

c. What is the profit when

43

units are produced and​ sold?

d. How many units must be sold to break even on this​ product?

a. Write the profit function.

P(x)=

​(Simplify your​ answer.)

Answer 1

Answer

a)

C(x) = 19,250 + 70x + x^{​​​​​​2}​​​

R(x) = 655x

P(x) = R(x) - C(x) [Profit = Revenue - Cost]

P(x) = 655x - (19,250 + 70x + x²)

P(x) = 655x - 19,250 - 70x - x²

P(x) = 585x - 19,250 - x²

So, Profit Function is P(x) = 585x - 19,250 - x²

b)

Profit when 25 units are produced and sold will be P(25)

P(x) = 585x - 19,250 - x²

P(25) = (585*25) - 19,250 - (25)²

P(25) = 14,625 - 19,250 - 625

P(25) = - $5,250

= - $5,250 ( loss)

So, when 25 units are Sold there will be a loss of $5,250

c)

Profit when 43 units are produced and sold will be P(43)

P(x) = 585x - 19,250 - x²

P(43) = (585*43) - 19,250 - (43)²

P(43) = 25,155 - 19,250 - 1,849

P(43) = $4,056

So, when 43 units are sold there will be a profit of $4,056

d)

Break even point

At break even point there is no profit or loss. At break even point Profit = 0.

So, P(x) = 0, where x is the number of units sold

So,

585x - 19,250 - x² = 0

x² - 585x + 19,250 = 0

Solving the equation using middle term split

x² - 550x - 35x + 19,250 = 0

x(x-550) - 35 (x-550) = 0

(x-35) (x-550) = 0

So, x = 35 or x = 550

Solving P(35)

P(35) = (585*35) - 19,250 - (35)²

P(35) = 20,475 - 19,250 - 1,225

P(35) = 0

Solving P(550)

P(550) = (585*550) - 19,250 - (550)²

P(550) = 321,750 - 19,250 - 302,500

P(550) = 0

So, the company breaks even at 35 units as well as at 550 units.

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