A certain company has fixed costs of $15,000 for product and variable costs give

Question

A certain company has fixed costs of $15,000 for product and variable costs given by 140 + 0.04x dollars per units, where x is the number units. The selling price of the product is given by 300 - 0.06x dollars per unit. a) Write the Revenue, Cost, and Profit Functions revenue = ____ Cost = ____ Profit = ____ b) How many units must be produced and sold to break even? Use algebra and show your work. Round appropriately. c) Find the level of sales that maximizes revenue. d) Find the (1) maximum profit, (2) the number of units that will maximize profits, and (3) the selling price that will maximize profits. Use algebra and show your work or no credit will be given. Round appropriately.

Explanation / Answer

Solution (a)

Cost function C(x) = 15000 + (140 +0 .04x) = 15000 +140x + 0.04x2

Revenue function R(x) = px = (300 – 0.06x) x = 300x – 0.06x2

Profit function P(x) = R(x)-C(x) = -0.1x2 +160x – 15000

Solution (b)

When will the company break-even?

Break even is when R=C

300x – 0.06x2 = 15000 + 140x + 0.04x2

Subtract 300x and Add 0.06x2 from both sides

0 = 15000 -160x + x2

0 = (x – 150) (x – 100)

So x=1500, 100

Solution (c)

R(x) = 300x – 0.06x2

Find the vertex

a = -0.06 b = 300

x= -b/2a = -300/2*(-0.06) = 2500

Plug in x=2500 to p = 300 – 0.06x.

300 – 0.06(2500) = 150

A price of $150 will maximize revenue.

Solution (d)

Use P(x) = -.1x2 +160x – 15000

Find the vertex

a = -.1 b = 160

x= -b/2a = 800

-0.1*(800)2+160(800) – 15000 = 49000

Must sell 800 units to earn a maximum profit of $49,000.

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