The Acme Widget Company has found that if widgets are priced at s 389, then 1000 will be sold. They have also found that for every increase of $ 10, there will be 600 fewer widgets sold. The marginal cost of widgets is $ 97.25. The fixed costs for the Acme Widget Company are $ 8000. lf x represents the price of a widget find the following in terms of x The number of widgets that will be sold: The revenue generated by the sale of widgets: The cost of producing just enough widgets to meet demand: The proft from selling widgets: Find the price that will maximize profits from the sale of widgets:

See the explanation below.

Explanation:

a. The number of widgets that will be sold

Let y represent the number of widgets that will be sold, and and we already have x as price of widget, we therefore have:

y - 1,000 = (-600/10) * (x - 389)

y - 1,000 = -60 * (x - 389)

y = 1,000 - [60 * (x - 389) ]

y = 1,000 - 60x + 23,340

y = 24,340 - 60x

b. The revenue generated by the sale of widgets

Let R represent Revenue, therefore we have:

R = xy

R = x(24,340 - 60x)

R = 24,340x - 60x²

c. The cost of producing just enough widgets to meet demand

Let C represent total cost, we therefore have:

C = 8,000 + 97.25y  

C = 8,000 + 97.25(24,340 - 60x)

C = 8,000 + 2,367,065  - 5,835x

C =  2,375,065  - 5,835x

d. The proft from selling widgets

Let P represent profit, we therefore have:

P = R - C

P = 24,340x - 60x²  - (2,375,065  - 5,835x)

P = 24,340x - 60x²  - 2,375,065  + 5,835x

P = - 60x²  + 30,175x - 2,375,065

e. Find the price that will maximize profits from the sale of widgets

Profit is optimum when dP/dx = 0

Therefore, we have

0 = - 120x + 30,175

120x = 30,175

x = 30,175/120 = $251.46