Question

Suppose that the cost of producing x items per hour is given by C(x) where C(x) = x^2+10, and the number of items sold per hour at a price of p dollars per item is x = 16 − p.
(a) Find in terms of x the revenue gained from the sales.

Answer 1

Okay, so if I tell you that revenue is \(px\), does it make sense to you?

Answer 2

\(p\) is the price, which is also the revenue per item. \(x\) is the number of items

Answer 3

yea

Answer 4

\[
x = 16 − p
\]Multiplying both sides by \(x\) get you:\[
x^2 = 16x-px
\]

Answer 5

And solving for \(px\):\[
x^2 = 16x - px \\
px + x^2 = 16x \\
px = 16x-x^2
\]

Answer 6

Does that make sense to you?

Answer 7

yep so far

Answer 8

\[
R(x) = px \implies R(x) = 16x-x^2
\]

Answer 9

The answer is already there

Answer 10

oh so the question didnt need x^2 + 10

Answer 11

Not part (a)

Answer 12

mind if i post the full question?

Answer 13

Go ahead

Answer 14

(b) Hence show that the profit
achieved per hour is given by −2x^2 +16x−10.
(c) Find the number of items that should
be produced each hour in order to maximise the profit.
(d) Find the maximum profit.
I can do part (d) so dw about that part

Answer 15

Okay, well total profit (or just profit) is just total cost minus total revenue

Answer 16

Wait...

Answer 17

What is the formula for profit?

Answer 18

px?

Answer 19

No, that was the formula for revenue.

Answer 20

px - c(x)

Answer 21

Okay, so then substitute in what you solved, and see what it simplifies to...

Answer 22

yea same as that part(b) gave us ^^

Answer 23

Okay so now for c

Answer 24

What is \(x\) such that \(−2x^2 +16x−10\) is maximized?

Answer 25

hmm?

Answer 26

4?

Answer 27

LOL, how did you get that?

Answer 28

x = -b/2a

Answer 29

Eh? Is that how you get the max? I've never seen that... Interesting.

Answer 30

x = 4 so max = 22

Answer 31

Let me think aloud here.... \[
ax^2+bx+c=f(x) \\
f'(x) = 0 \implies 2ax+b = 0 \implies x = -\frac{b}{2a}
\]Oh wow, you're right, good job.

Answer 32

ok anyways thanks for your help again ^^

Answer 33

Interesting formula though.

Answer 34

In Australia it is a formula you need to memorise.