Question

The profit f(x) of manufacturing x widgets is given by f(x)=-500+250x-0.25x^2

Show how you obtained each answer

a. What number of widgets produces a maximum profit

b.what is the maximum profit

c.To achieve sales goals (and be eligible for bonuses) the company wins for $50,000 in sales. How many widgets must they sell to earn bonuses

Answer 1

(a) 500 units

(b) $62000

(c) 281 ≤ x ≤ 719 units

Explanation:

the maximum profit is at the extreme point of f(x), where the 1st derivative = 0

`f(x)= -500 + 250 x - 0.25x^2`

`f'(x)=0`

`250-0.25(2)x^((2-1))=0`

`250-0.5x=0`
`x=500`

(a)

x = 500 units

(b)

`f(500)=-500+250(500)-0.25(500)^2`

`=\$62000`

(c)

`f(x)=50000`

`-500 + 250 x - 0.25x^2=50000`

`0.25x^2-250x+50500=0`

`x=(-b\pm√(b^2-4ac) )/(2a)`

`=(-(-250)\pm√((-250)^2-4(0.25)(50500)) )/(2(0.25))`

`=(250\pm109.54 )/(0.5)`

`=281\ or\ 719\ units`

Therefore, total units is between and inclusive of 281 and 719 units