Function Problem: cost $20/unit, q=120-p sold at price $p

[jemini79]

This is my first math class in two years so I have lots of questions but my question tonight is regarding a word problem. I have the answer but reviewing the chapter I can't find anything that speaks to this type of question so I don't know where the answer came from.

Problem:
A manufacturer can produce tape recorders at a cost of $20 apiece. It is estimated that if the tape recorders are sold for p dollars apiece, consumers will buy q=120-p recorders each month.
a) Express the manufacturer's profit P as a function of q. The book gives the answer: (120-q)q-20q
What I need help with is understanding how the book came up with that equation. Any help would be much appreciated.

 

[Loren]

Re: Function Problem

>The book gives the answer: (120-q)q-20q.

Is there a typo?

 

[jemini79]

Re: Function Problem

(120-q)q-20q is what is posted in the back of the book. There might be a typo and maybe that is why I am confused. In the question it states that q=120-p so I would have thought that p would have been a part of the calculation. Since the question asks to express profit as a function of q maybe p was switched out for q. I am not sure.

 

[Deleted member 4993]

Re: Function Problem

This is my first math class in two years so I have lots of questions but my question tonight is regarding a word problem. I have the answer but reviewing the chapter I can't find anything that speaks to this type of question so I don't know where the answer came from.
Problem:
A manufacturer can produce tape recorders at a cost of $20 apiece. It is estimated that if the tape recorders are sold for p dollars apiece, consumers will buy q=120-p recorders each month.
a) Express the manufacturer's profit P as a function of q. The book gives the answer: (120-q)q-20q
What I need help with is understanding how the book came up with that equation. Any help would be much appreciated.

Forget the answer in the book - just do it as if that given answer does not exist.

If you cannot even start:

Please tell us the definitions of profit.

What is the cost of manufacturing 'q' units?

What is the total revenue after selling 'q' units?

What is the profit after selling 'q' units?

This problem is more of understanding the definitions - than actusl arithmatic.

Please show us your work, indicating exactly where you are stuck - so that we know where to begin to help you.

 

[Loren]

Re: Function Problem

I believe the answer in the answer book is correct.
Notice that...
(120-q)q-20q = 120q-q[sup:19rvkotw]2[/sup:19rvkotw]-20q = 100q-q[sup:19rvkotw]2[/sup:19rvkotw] = q(100-q) which is a much better/simplified answer.

 

[soroban]

Re: Function Problem

Hello, jemini79!

I had to baby-talk my through it . . .

A manufacturer can produce tape recorders at a cost of $20 apiece.
It is estimated that if the recorders are sold for \(\displaystyle p\) dollars each,
consumers will buy \(\displaystyle q\:=\:120-p\) recorders each month.

a) Express the manufacturer's profit \(\displaystyle P\) as a function of \(\displaystyle q.\)

The book gives the answer: .\(\displaystyle P \:=\: (120-q)q-20q\)

We know that: .\(\displaystyle \text{Profit} \:=\:\text{Revenue} - \text{Cost}\) .[1]


Suppose they manufacture and sell \(\displaystyle q\) units.

They sell \(\displaystyle q\) units at \(\displaystyle p\) dollars each.
. . The revenue (income) is: .\(\displaystyle \text{Revenue} \:=\q\) .[2]

. . We're told that: .\(\displaystyle q \:=\:120-p \quad\Rightarrow\quad p \:=\:120-q\)

. . Substitute into [2]: .\(\displaystyle \text{Revenue} \;=\;(120-q)q\) .[3]


They manufacture \(\displaystyle q\) units at a cost of $20 each.
. . Their cost is: .\(\displaystyle \text{Cost} \:=\:20q\) .[4]


Substitute [3] and [4] into [1]:

. . \(\displaystyle \text{Profit} \;=\;\underbrace{\text{Revenue}}_{(120-q)q} - \underbrace{\text{Cost}}_{20q}\)


Got it?