Learning Curve

Dear Mr Moffat,

Could you kindly help with below question.

The Q organization develops a new products that are in high demand.
Market research has discovered that the price demand relationship for n anew DVD prduct during the initial launch phase will be as follows:

Price Demand
$ Units

100 10000
80 20000
69 30000
62 40000

Production of the DVD recorder would occur in batches of 10,000 units and the production director believes that 50% of the variable manufacturing cost would be affected by a learning curve. This would apply to each batch produced and continue at a constant rate of learning up to a production volume of 40,000 units when the learning would be complete. The production director estimates that the unit variable manufacturing cost of the first batch would be $60($30 of which is subject to the effect of the learning curve, and $30 of which is unaffected), whereas the average unit variable manufacturing cost of all four batches would be $52.71

Require

a) Calculate the rate of leaning that is expected by the production director.
b) Calculate the optimum price at which Q should sell the DVD recorder in order to maximise its profit during the initial launch.

Answer provided:

Cumulative n. Of unit T v.c. Affected by L Not effected by L
10,000 60 30 30
20,000 56,10 26,10 30
30,000 54,06 24,06 30
40,000 52,71 22,71 30

Calculate the rate of learning

At 10,000 units variable cost affected by learning = $30
At 40,000 units variable cost affected by learning = 22,71

Let the rate of learning be R
22.71/30 0 r2
r2= 0.757
r = 0.87

(Note. Also using the average costs per unit for 10,000 and 20,000 units 26.10/30= r therefore r is 0.87) This note was provided in the answer, but I still don t understand where 26,10 come from

Variable cost affected by learning for 20.000 units
30r = 30*0,87=26,10

To calculate the variable cost affected by learning for 30.000 units requires the use of a formula for learning curve
30*3-0,201 ( log0,87/log2)
=24.06

Could you please explain the answer of this question? I listened to the lecture and I didn t have any problem to answer other questions about the learning curve, but I don’ t understand the logic behind the working provided.

Thanks in advance for your help
Gabriella