Question

A manufacturer can produce tape recorders at a cost of $25 apiece. It is estimated that if the tape recorders are sold for p dollars apiece, consumers will buy q=115-p recorders each month. At what rate is profit changing when q=15 recorders are produced? Is the profit increasing or decreasing at this level of production?

I need help so bad!!!
Thanks!

Answer 1

maybe we can do this

Answer 2

you sell \(115-p\) for \(p\) dollars, making a total of \(p(115-p)\) dollars (the amount you sell them for times the number you sell
but each of the \(115-p\) you sell cost you $25 so your profit is the difference
\[p(115-p)-25p\]

Answer 3

multiply this out, and you get a profit of
\[p(115-p)-25p=90p-p^2\]

Answer 4

The answer was $60 increasing, but I am not sure how they did it, to get that answer. :(

Answer 5

@rocal2 was this question open overnight? Just saw it bumped up...

Answer 6

Yes, I would like a good explaination.. Of why they got 60 Incresing and I got $65.

Answer 7

You need a good function for profit...

Profit = sales price - manufacturing cost (at least for this example)
P is profit function, p = sales price, q = # of units, and cost is given at 25

so P(p) = pq - 25q because you receive p times q revenue for q units sold at p dollars
and your manufacturing costs are 25 times q

P(p) = p(q-25)

Answer 8

So, you can express profit if you know the sales price, p, and number of units, q

Answer 9

hm... hold on, need to re-read the problem. I didn't notice at first that q is also a function of p

Answer 10

Ok, back on track.
P(p) = pq - 25q <<-- profit is price x quantity minus manufacturing cost x quantity
= p(115-p) - 25(115-p) <<-- sub in expression for q in terms of p
P(p) = 115p - p^2 - 2875 + 25p
P(p) = -p^2 +140p - 2875

The rate of change of P(p) is P ' (p) = -2p + 140

Evaluate rate of change of profit when q = 15.... q=15 -->> q = 115-p -->> p = 100

So P'(p) at q=15 which is p=100 means P'(100) = -2(100) +140 = -60

Profit is changing at a rate of -60 when customers buy 15 units.

I am confused though... I get this to be -60, which I would think is "decreasing". I think the value of 60 is correct, but if the correct answer is supposed to be increasing, we need to go back and find a place where things got flipped.

Answer 11

Thank you so much, I will keep this example. And I'll check why they said increasing... You are the best... :)

Answer 12

I haven't solved it yet, but the problem is that you need a profit function in terms of q, not p
And rate of change of profit is really rate of change of profit WITH RESPECT TO q

Here's a start... same idea, but I subbed in for every p as (115-q)

P(p) = -p^2 +140p - 2875
P(q) = -(115-q)^2 + 140(115-q) - 2875
P(q) = -( 13225 - 2(115)q + q^2) + 16100 - 140q - 2875

Answer 13

Simplify that down, take the first derivative to get P ' (q), and then sub in q=15. I bet you will get P ' (q=15) = 60 (increasing)

Answer 14

Here's a quick plot of the P(q) curve, and you can plainly see that at q = 15 (left side of down-facing parabola), P(q) is a positive slope, or increasing.

http://www.wolframalpha.com/input/?i=P%28q%29+%3D+-%28115-q%29^2+%2B+140%28115-q%29+-+2875

Answer 15

Ok, I understand I need to practice this one more to be clear.. I appreciate.

Answer 16

good luck... if I had just realized it needed to have been profit as a function of quantity instead of profit as a function of price, we would have arrived at the right solution much sooner... sorry for the detour, but hope it helped a little. Practice sounds like a good idea (for me too!!)

Answer 17

I really appreciate, this mean a lot to me. I have quiz and this will be on it. So I have to do it good.. Thanks so much, I hope I can count on you next time! lol! ;)