Mosaic Industries is an international company that sells cell phone cases. Their market research predicts that the average number of cases they will sell each month is modelled by the equation

\(n = 20000 – 1000x\)

where \(x\) represents the selling price, in euro (EUR), of each case. A salesperson suggests that Mosaic Industries should sell the cases for 16 EUR each.

(a) Find the average number of cases that this model predicts Mosaic Industries will sell at this price. [2]

(b) Calculate Mosaic Industries’ average monthly income, before any expenses, at this selling price. [2]

(c) Hence, write down the function \(R(x)\) that can be used to predict Mosaic Industries’ average monthly income, before expenses, at any selling price, \(x\). [1]

Mosaic Industries has 10 000 EUR of fixed monthly operational costs. Additionally, Mosaic Industries must pay their phone case supplier 10 EUR for each case.

(d) Calculate Mosaic Industries’ average monthly profit if they sell each case at a price of 16 EUR. [3]

(e) Show that the average monthly profit for any selling price, \(x\), can be found using the function \(P(x) = -1000x^2 + 30000x – 210000\). [2]

(f) (i) Find \(P'(x)\).
(ii) Show that the salesperson’s selling price does not maximize their average monthly profit. [4]

Mosaic Industries negotiates a new deal with their phone case supplier. Under the new deal, the supplier agrees to discount the cost of each case based on the number of cases purchased by Mosaic Industries. The cost charged by the supplier for each case can be found using the function

\(C(n) = 10 – 0.0001n\)

where \(n\) represents the number of cases sold by Mosaic Industries.

(g) Find the function that can be used to find Mosaic Industries’ average monthly profit using the new deal from the supplier. [3]

(h) Hence, find the selling price, per case, that Mosaic Industries should choose in order to maximize their average monthly profit under the new deal. [2]