A profit function is derived from the production cost and revenue function for a given item.?

Method 1 : Completing square

27a.
P(x) = -0.15x² + 900x - 120500
P(x) = -0.15(x² - 6000x) - 120500
P(x) = -0.15(x² - 2*3000x + 3000²) + 0.15*3000² - 120500
P(x) = -0.15(x - 3000)² + 1229500

For all real values of x, -0.15(x - 3000)² ≤ 0
Then, P(x) = -0.15(x - 3000)² + 1229500 ≤ 1229500
Maximum P(x) when x = 3000

Answer = 3000 units

27b.
When x = 3000, P(x) = 1229500

Answer = $1,229,500,000


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Method 2 : Differentiation

27a.
P(x) = -0.15x² + 900x - 120500
P'(x) = -0.30x + 900 = -0.30(x - 3000)
P"(x) = -0.30

When x = 3000 :
P'(x) = 0 and P"(x) = -0.30 < 0
Hence, maximum P(x) when x = 3000

Answer : 3000 units

27b.
When x = 3000 :
P(3000) = -0.15(3000)² + 900(3000) - 120500 = 1229500

Answer = $1,229,500,000

I don't know if you know calculus or not, so I'll do this with algebra (and may show the calculus at the end).

You have this as your profit function:

P(x) = -0.15x² + 900x - 120500

If we put this into vertex form:

P(x) = a(x - h) + k

Then "h" will be the number of units you have to sell to maximize the profit, and "k" will be that profit (answering both questions at once).

So to do that, we need to complete the square, which means we have to have the right side in the form of (x² + bx). So add the constant term to both sides and divide both sides by -0.15:

P(x) = -0.15x² + 900x - 120500
P(x) + 120500 = -0.15x² + 900x
-( P(x) + 120500) / 0.15 = x² - 6000x

Now to complete the square, take x's coefficeint (-6000), half it (-3000), then square it (9000000). Add that to both sides:

9000000 - ( P(x) + 120500) / 0.15 = x² - 6000x + 9000000

Now you can factor the right side as a perfect square trinomial:

9000000 - ( P(x) + 120500) / 0.15 = (x - 3000)²

Now solve for P(x) again keeping that binomial square in tact. Subtract 9000000 from both sides:

-( P(x) + 120500) / 0.15 = (x - 3000)² - 9000000

Multiply both sides by -0.15:

P(x) + 120500 = -0.15(x - 3000)² + 1350000

And finally subtract 120500 from both sides:

P(x) = -0.15(x - 3000)² + 1229500

So the maximum profit of $1,229,500 can be made by selling 3000 units.

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To show the calculus version, we solve for the zero of the first derivative, so:

P(x) = -0.15x² + 900x - 120500
P'(x) = -0.3x + 900
0 = -0.3x + 900
0.3x = 900
x = 3000

And we get the same answer for part a as we did the first time. Now solve for P(3000) for part B:

P(x) = -0.15x² + 900x - 120500
P(3000) = -0.15(3000)² + 900(3000) - 120500
P(3000) = -0.15(9000000) + 900(3000) - 120500
P(3000) = -1350000 + 2700000 - 120500
P(3000) = $1,229,500

and we get the same answer as above for part B. So I showed two ways for you to do this problem, depending on if you know calculus or only algebra.

Best answer if this helped, please.

Any time a question asks you about maximums or minimums, you know you'll need to put the equation into vertex form. Here, we'll complete the square.

p(x) = -0.15x^2 + 900x - 120 500

p(x) = -0.15 (x^2 - 6000) - 120 500

p(x) = -0.15 (x^2 - 6000 + (6000/2)^2 - (6000/2)^2) - 120 500

p(x) = -0.15 (x^2 - 6000 + 3000^2) - 0.15(-3000^2) - 120 500

p(x) = -0.15 (x - 3000)(x - 3000) - 0.15(-9 000 000) - 120 500

p(x) = -0.15 (x - 3000)^2 + 1 350 000 - 120 500

p(x) = -0.15 (x - 3000)^2 + 1 229 500

y = a(x-h)^2 + k is vertex form, where k is your max/min, and -h is the value at which that occurs.

So in other words, your max profit is $1 229 500, and you need to sell at least 3000 units to achieve that.

a)
P(x) = -0.15x^2+900x-120,500
P'(x) = (-0.15)(2x) + 900 = 0
-0.30x = -900
x = 900/0.30
x = 3,000 units must be produced to maximize the proit

d^2P/dx^2 = -0.30 < 0 confirms this

b)
Maximum proit = P(3000)
= -0.15(3000)^2 +900(3000) -120500
= 1,229,500 dollars

A graphing calculator can answer both questions easily. See the source link.

a) 3000 units must be sold to maximize profit

b) Maximum profit is $1,229,500

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For quadratic ax^2 +bx +c, the vertex is located at x = -b/(2a). Here, that is -900/(2*(-0.15)) = 900/.3 = 3000 . . . units

Then the maximup profit is
.. P(3000) = (-.15*3000 +900)*3000 -120500
.. = 450*3000 -120500
.. = 1,229,500 . . . dollars