CONTEXT

A champagne trader wants to offer a bundle to the 200 employees of a company. It is supplied from a winemaker at a cost of 30 per bottle, knowing that a discount proportional to the volume delivered applies. Direct sale, a bottle of champagne of the same category can be bought at 40 .

CALCULATION OF TURNOVER

1) a) We denote x the selling price in . What is the maximum price that can be applied to make the offer attractive to the market?

Maximum price = ..

On the spreadsheet, complete the Price column up to this maximum value, in increments of $ 1.

1) b) Call N (x) the demand (i.e. the number of employees interested) at price x. What can be expected as a request for a price of 0 ? from 40 ?

N (0) = N (40) = .

1) c) We assume here that N (x) = a.x + b. Using the previous question, determine a and b:

N (x) = . x + .

On the spreadsheet, use this formula to complete the Request column.

2) The income R (x) = x.N (x) corresponds to the turnover achieved. Determine the expression of R (x) as a quadratic function:

R (x) = . ..

On the spreadsheet, complete the Income column.

CALCULATION OF LOADS

3) a) We apply a reduction coefficient r proportional to the volume, so that the unit cost per bottle is given by the formula U (x) = 30 - r.N (x)

On the spreadsheet to the right of the Reduction box, enter the r value given to you by the teacher

Express U (x) in terms of x:

U (x) = .

On the spreadsheet, complete the Unit column.

3) b) The charges C (x) = U (x) .N (x) correspond to the payment of the products to the supplier. Express C (x) as a function of x as a quadratic function:

C (x) = ..

On the spreadsheet, then complete the Costs column.

SEARCH FOR THE OPTIMUM BENEFIT

4) a) Profit B (x) corresponds to the differential between income and expenses. Express B (x) as a function of x as a quadratic function:

B (x) = ...

On the spreadsheet, then fill in the column Profit B (x)

4) b) Determine the selling price (to the nearest euro) corresponding to the maximum profit:

Optimal price: . Maximum benefit: .

Add a chart (with title and legend on the axes) showing profit versus price x.

4) c) Find the exact values of the optimal price (to the nearest cent) and of the maximum profit (to the nearest euro) by studying the sign of the derivative B and the variations of B on [0; 40]: