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Studying for a midterm.

The demand function for a manufacture's product is $p=1000-\frac1{80} q$

Where $p$ is the price (in dollars) per unit when $q$ units are demanded (per week) by consumers. Answer the following questions.

1) Write the Revenue function $R(q)$ in terms of $q$.

2) Find the level of production that will maximize revenue.

3)Suppose there is a fi xed cost of $174500, to set up the manufacture and a producing cost of 125 dollars per unit. Find the break even quantities.

First: To find the revenue function.

I know that Revenue=$p*q$ so:

$$R(q)=p*q$$

$$p=1000-\frac1{80}q$$

$$R(q)=(1000-\frac1{80}q)*q$$

$$=1000q-\frac1{80}q^2$$

I believe this is right.

Now to find the level of production to maxime revenue we must find the first derivative of the revenue function.

$$R'(q)=1000-2(\frac1{80}q)$$ $$2(\frac1{80}q)=1000$$ $$\frac1{80}q=500$$ $$q=40000$$ Input this into our demand function:

$$p=1000-\frac1{80}40000$$ $$p=500$$

Now I don't know if this is right, please correct me if I'm wrong.

Now I'm not sure how to find the break even quantities, I would appreciate help, at least to get me started.

Cheers.

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  • $\begingroup$ For the breakeven point, you will have to solve a quadratic, as mentioned in an answer. Probably you will use the Quadratic Formula, though in fact one can factor. There will be two positive roots. The smaller one is the breakeven point. It will be a very "round" number. $\endgroup$ – André Nicolas Oct 10 '13 at 21:57
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    $\begingroup$ How are there 60k views but only 4 upvotes...... $\endgroup$ – Tdonut Nov 9 '15 at 22:02
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Your work is correct. To find the break even quantities, you need to find where the Revenue function is equal to the cost function. Your cost function is $C(q)=174500+125q$.

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  • $\begingroup$ I gave it a shot and I end up with $0=q^2-70000q+1396000$, although this is unfactorable. $\endgroup$ – Unknown Oct 10 '13 at 22:08
  • $\begingroup$ @Unknown You don't need to factor. The quadratic formula will give you your answers. $\endgroup$ – Joe Johnson 126 Oct 10 '13 at 22:08
  • $\begingroup$ I'm afraid I don't quite understand, can you be more specific? $\endgroup$ – Unknown Oct 10 '13 at 22:11
  • $\begingroup$ Do I just input 40000 in place of q? $\endgroup$ – Unknown Oct 10 '13 at 22:14
  • $\begingroup$ @Unknown The quadratic formula is a way to solve $0=q^2-70000q+1396000$. $\endgroup$ – Joe Johnson 126 Oct 10 '13 at 22:15

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