Is it possible for the company to make a profit 15,000 ?

Question

For a certain company, the cost for producing x items is 55x+300 and the revenue for selling x items is 95x−0.5x^2.

Find two values of x that will create a profit of $300.

Answer 1

Sales-costs=0.

Solve.

Answer 2

We will use the basic formula for calculating profit as

Profit (P) = Revenue (R) minus Costs (C) - the money you get minus the money you spend to get it is your profit. If something costs $10 to make and you sell it for $15, your profit is $15-$10 = $5.

In this case, both revenue R and cost C are functions of x. Therefore, we can write in the function notation, as follows:

R(x) = 95x-0.5x²

C(x) = 55x+300

Therefore, profit P(x) = R(x) - C(x) = (95x-0.5x²) - (55x+300) = -0.5x² + 40x - 300

And we are told that for some x, this profit reaches $300. Therefore,

-0.5x² +40x - 300 = 300

Multiplying both sides by (-1), we get

0.5x² -40x + 300 = -300

Or, adding 300 on both sides,

0.5x² -40x + 600 = 0

Multiplying both sides by 2, we get

x² -80x + 1200 = 0

Factoring the above, the numbers 60 and 20 seem to fit the bill. 60*20 = 1200 (constant term), 60+20 = 80 (middle term)

So, rearranging the above, we get

x² -60x - 20x + 60*20 = 0

Factoring the above, we get

x(x-60)-20(x-60) = 0

or (x-60)(x-20) = 0

Therefore either x=60 or x=20 will result in a profit of $300

CHECK:

Use x=60

P(60) = -0.5(60)²+40(60)-300 = -0.5*3600+2400-300 = -1800+2400-300 = 2400-2100 = $300

Use x = 20

P(20) = -0.5(20)²+40(20)-300 = -0.5*400+800-300 = -200+800-300 = 800-500 = $300