Question

In order to improve the financial Situation of Thoughts, the Manager looked into expansion plans and decided to offer caterings for events. For the catering business, the manager is offering his customers a selling price p that depends on demand, where p= 9 - 0.02*D.

a- Calculate the Quantity that Maximises Revenue.

b- Calculate the Quantity that Maximises Profit (Refer to problem 1 for costs).

Problem 1 Costs:

Rental $80 per day

Coffee beans $5 per Coffee Cup

Sugar $0.3 per Coffee Cup

Flavors $ 0.5 per Coffee Cup

Filtered water $0.2 per Coffee Cup

Labor $30 per day

c- Calculate the Breakeven point(s).

No Ready-to use Formula should be used in this Problem, show your iterations. Ready-to-use formulas and Calculator-derived results will NOT be accepted

Answer 1

a) p = 9 - 002D

Total Revenue TR = p*D = (9 - 0.02 * D) * D = 9D - 0.02 D^2

In order to maximize total revenue,

d/dD (TR) = 0

or d/dD (9D - 0.02 D^2) = 0

or 9 - 0.04 D = 0

or D = 9/0.04 = 225

So the revenue maximizing quantity is 225

Maximum Revenue = 9D - 0.02 D^2 = 9 * 225 - 0.02 * 225^2 = 2025 - 1012.5 = 1012.5

b) Let D cups per day maximize profit.

Total Revenue from selling D cups = TR = p*D = (9 - 0.02 * D) * D = 9D - 0.02 D^2

Total Cost = TC = Rental / Day + Labour / Day + Coffee Beans / Day + Sugar / Day + Flavors/ day + Filtered Water / Day

= 80 + 30 + 5D + 0.3D + 0.5D + 0.2D

= 110 + 6D

Total Profit = TP = TR - TC = 9D - 0.02 D^2 - (110 + 6D) = 3D - 0.02 * D^2 - 110

To maximize total profit, d/dD (TP) = 0

or d/dD (3D - 0.02 * D^2 - 110) = 0

or 3 - 0.04D = 0

or D = 3/0.04 = 75

So the profit maximizing quantity is 75.

The total profit will be TP = 3D - 0.02 * D^2 - 110 = 3 * 75 - 0.02 * 75^2 - 110 = 225 - 112.5 - 110 = $2.5

c) At breakeven point, TP = 0

or 3D - 0.02 * D^2 - 110 = 0

or D^2 - 150D + 5500 = 0

or D = ( -(-150) +/- sqrt ( (-150)^2- 4 * 1 * 5500) ) / 2

or D = (150 +/- sqrt (22500 - 22000) ) * 0.5

or D = (150 +/- sqrt (500)) * 0.5

or D = (150 +/- 22.36) * 0.5

or D = 172.36 / 2 or 127.64 / 2

D = 86.18 or 63.82

Since quantity cannot be fractional, so the nearest breakeven points will be quantity of 64 or 86.

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