Learning Rate. Qn 2 BPP Revisons Kit Mock 2 Pg 315

Dear Tutor
I came across this question in the BPP revision kit and has given me very hard time..,

…….Production of DVD occur in batches of 10,000 units the P director believes that 50% of variable manufacturing cost would be affected by a learning curve. This would apply to each batch produced & continue at a constant rate of learning up to a production volume of 40,000 units when a learning would be complete. The p Director estimates that the variable manufacturing cost of the first batch would be $60($30 of which is subject to the effect of learning curve, and $30 of which is unaffected), whereas the average unit variable manufacturing cost of all four batches would be $52.71. there are no non-manufacturing variable costs associated with the DVD recorder.

Calculate the rate of learning that is expected by the production director?

In their solution…..,
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 = r2
r2 = 0.757
r = 0.87

Why should r be squared?, and why 22.71/30…infact the whole reasoning has defeated my understanding….

Please help out. ( Modesto from kampala,Uganda)