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 fixed 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.
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 $C(q)=174500+125q$.