# Question #952b7

May 7, 2017

Level of production: ${q}_{0} = 200$
Total optimal revenue : ${P}_{0} = 240000$

#### Explanation:

Since we will need the totale revenue, it is the price of the whole order, hence price per unit times number of unit:

$P \left(q\right) = q p \left(q\right) = 2400 q - 6 {q}^{2}$

We need to find the maximum of this quadratic equation, and we know that quadratic equations have only one stationary point which is a maximum if the leading term is negative.

Hence let's find this stationary point, which is the only point ${q}_{0}$: such that

$P ' \left({q}_{0}\right) = 0$

Hence

$P ' \left(q\right) = 2400 - 12 q = 0 R i g h t a r r o w {q}_{0} = 200$

Hence its total revenue will be

${P}_{0} = P \left({q}_{0}\right) = 2400 {q}_{0} - 6 {q}_{0}^{2} = 480000 - 240000 = 240000$