I am having some trouble with this question in my grade 11 math textbook. We are learning applications in differential calculus. It is a question that can be solved many different ways but our teacher is asking us to do it using calculus. I am stuck and do not know what to do here
The question is:
The total cost of producing $x$ blankets per day is $1/4 (x^2) + 8x+20$ dollars, and for this production level each blanket may be sold for $(23 – 1/2 (x))$ dollars. How many blankets should be produced per day to maximise the total profit?
If you could explain in detail how you arrived at the answer (which is 10 blankets) using differential calculus it would be great.
P.S I know that to find profit you need to subtract revenue with cost, but I do not know how to get there.
Best Answer
If each blanket may be sold for $(23−\frac{1}{2}x)$ dollars, the profit for $x$ produced blankets is: $$ p(x)=x\left(23−\frac{1}{2}x \right)-\left(\frac{1}{4}x^2+8x-20\right)=-\frac{3}{4}x^2+15x-20 $$ Now derive:
$$ p'(x)=-\frac{3}{2}x+15 $$
and find the maximum for $p'(x)=0 \Rightarrow x=10$.