A Study of the Costs to Produce Airplanes in World $M C(x)=M M_{0} x^{\log _{2} 0.9}$

question 45

Short Answer

A study of the costs to produce airplanes in World War II led to the theory of "learning curves," the idea of which is that the marginal cost per plane decreases over the duration of a production run.In other words, with experience, staff on an assembly line can produce planes with greater efficiency.The 90% learning curve describes a typical situation where the marginal cost, MC, to produce the x^th plane is given by $M C(x)=M M_{0} x^{\log _{2} 0.9}$ , where $M_{0}$ = marginal cost to produce the first plane.[Note: You may use the fact that $\log _{2} x=\frac{\ln x}{\ln 2}$ .] Recall that marginal cost is related to total cost as follows: $M C(x)=C^{\prime}(x)$ , where C(x)= total cost to produce x units.
Given this, and the formula for MC(x)with $M_{0}$ = $500,000, find a formula for C(x).If the constant for C(x)is $20 million, what is C(50), the cost of producing 50 planes, to the nearest thousand dollars?

---

Definitions:

Initial Investment

The initial amount of money invested in a project or business venture.

CCA Rate

Refers to Capital Cost Allowance rate, which is the rate at which a business can claim tax depreciation on certain properties or equipment in Canada.

CCA Class

Canadian Capital Cost Allowance Class; a categorization in Canadian tax law that determines the depreciation rate for tax purposes on different types of assets.

Required Rate

The lowest expected gain an investor aims to receive from an investment in a specific asset, taking into account the associated risk.