Question

"A contributing factor to an airplane's duel 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 3,800 hours per
year and will average 560 miles per hour. The amount of fuel that
the airplane consumes can be expressed as:

z = 0.0469 - (7.94*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 $4.33 per gallon.

The initial cost of the engine system as a function of the bypass
ratio is $267,000 + $2,600y^2.

The engine system will be used for 13 years. At the end of 13
years, the salvage value of the engine system as a function of
bypass ratio is $7,000y. 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?

(The optimal answer for the bypass ratio is between 4 to 12, but it
should not be necessary to consider that constraint in your
calculations.)"

Answer #1

Initial cost of engine (**IE**) = 267,000 +
2,600y^{2}

Salvage value (**SV**) = 7000y

Amount of fuel consumed in a year = 3800 x 560 x 0.0469 - (7.94
x 10^{-4}) y

Cost of fuel per year (**AF**)= 4.33 x 3800 x 560 x
[0.0469 - (7.94 x 10^{-4}) y]

It is required to minimize **IE - SV + AF** =>
267000 + 2,600y^{2} - 7000y + 9214240 [0.0469 - (7.94 x
10^{-4} ) y]

= 267000 + 2600y^{2} - 7000y +
432147.85 - 7316.10y

= 2600y^{2} - 14316.10y +
699147.85

Taking derivative of the expression and setting it equal to zero will minimize the value of the expression at desired y.

5200y - 14316.10 = 0

**y = 2.75**

"A contributing factor to an airplane's duel 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 3,200 hours per
year and will average 450 miles per hour. The amount of fuel that
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