Question

A company’s total cost from manufacturing and selling x units of their product is given by: y = 2x2 – 600x + 49,000. How many units should be manufactured in order to minimize cost, and what is the minimum cost? a. (100, $9,000)b. (125, $3,250)c. (150, $4,000)d. (170, $4,800)e. (200, $9,000) The cost of producing x hundred items is given by the equation C(x) = 2x2 – 4x + 4 and the revenue generated from sales of x hundred units is given by the equation R(x) = –4x2 + 74x – 176. What values of x will the company break even? a. 2 or 12b. 2 or 16c. 3 or 8d. 3 or 10e. 4 or 12 The profit from manufacturing and selling x units of a tablet PC is given by:P(x) = –.03x2 + 1,200x – 34,000How much profit should the company expect from selling 20,000 tablets? a. $36,000,000b. $24,000,000c. $23,966,000 d. $22,000,000 e. $11,966,000

Answer 1

a) c ( 150, $ 4,000)

b) d ( 3 or 10 )

c) e 11,966,000 $

Explanation:

a) C(t) = 2x² - 600x + 49000

Taking derivatives on both sides of the equation

C´(t) = 4x - 600 C´(t) = 0 4x - 600 = 0 x = 150

And minimun cost is:

C(min) = 2x² - 600x + 49000 ⇒ C(min) = 2* ( 150)² - 600* (150) +49000

C(min) = 4000 $

b) x hundred of tems cost

C(x) = 2x² - 4x + 4 and revenue generated is R(x) = -4x² + 74x -176

Then C(x) should be equal to R(x)

2x² - 4x + 4 = -4x² + 74x -176

Solving for x

6x² - 78x + 180 = 0 ⇒ x² - 13x + 30 = 0

x₁,₂ = 13 ± √( 169 - 120) /2 x₁ = 10 x₂ = 3

c) Th profit for manufacturing and selling tablets

P(x) - 0,03*x² + 1200x - 34000

where x is numbers of tablets. If the company sells 20000 tablets then

P(x) = - 0,03* (20000)² + 1200 (20000) - 34000

P(x) = - 0,03 * 4 *10⁸ + 24* 10⁶ - 34000

P(x) = - 12000000 + 24000000 - 34000

P(x) = 11966000 $