A contributing factor to an airplane\'s fuel consumption is the bypass ratio of

Question

A contributing factor to an airplane's fuel consumption is the bypass ratio of the engine system. The bypass ratio is the amount of air passing around the engine core relative to the amount of air passing through the core. An airplane manufacturer is designing a new airplane and wants to determine the bypass ratio for the airplane's engine system. The airplane will fly 3,700 hours per year and will average 500 miles per hour. The amount of fuel that the airplane consumes can be expressed as:
z = 0.0511 - (8.35*10^-4) * y
for 4 < y < 12
where y is the bypass ratio (a unitless number) and z is the number of gallons of fuel consumed per mile flown by the airplane. The cost of fuel remains constant at $4.12 per gallon.
The initial cost of the engine system as a function of the bypass ratio is $278,000 + $2,500y^2.
The engine system will be used for 15 years. At the end of 15 years, the salvage value of the engine system as a function of bypass ratio is $10,000y. The airplane manufacturer wants to minimize the annual equivalent cost (AEC) of the engine system (which includes the initial cost, the annual cost of fuel, and the salvage value). The manufacturer's MARR is 13.1%. What is the optimal bypass ratio rounded to the nearest tenth that minimizes the AEC of the engine system?

Explanation / Answer

Initial cost of engine (IE)= 278000 + 2500y2

Salvage value (SV)= 10000y

Amount of fuel consumed in a year = 3700*500* (0.0511 - (8.35*10-4 )y

Cost of fuel per year (AF)= 4.12*3700*500* (0.0511 - (8.35*10-4 )y)

We have to minimize IE-SV+AF= 278000 + 2500y2 - 10000y + 7622000(0.0511 - (8.35*10-4 )y)

= 278000 + 2500y2 - 10000y + 389484.2 - 6364.37y

= 2500y2 - 16364.37y + 667484.2

Taking derivative of the expression and putting it equal to zero will give us the minimum value of the expression at desired y.

5000y - 16364.37= 0

y = 3.27

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