Math Problem Statement
An airplane crosses the Atlantic Ocean (3000 miles) with an airspeed of 600 miles per hour. The cost C (in dollars) per passenger is given by C(x)equals150plusStartFraction x Over 15 EndFraction plusStartFraction 34 comma 000 Over x EndFraction , where x is the ground speed (airspeed plus or minus wind).
Solution
The cost function for the airplane flight per passenger is given as:
where:
- is the ground speed, which is the airspeed plus or minus the wind speed (measured in miles per hour),
- 150 is a fixed cost,
- is a cost component that increases with ground speed, and
- is a cost component that decreases as the ground speed increases.
Further Exploration
- How would the cost function change if the airspeed was different?
- At what ground speed is the cost per passenger minimized?
- How does wind speed affect the ground speed and thus the cost?
- How would the cost function change if the distance traveled was not 3,000 miles but a different value?
- Can we find the derivative of the cost function to analyze how changes with respect to ?
Tip:
Ground speed is influenced by both the airspeed of the plane and the wind conditions, which can either increase or decrease ground speed depending on whether it's a tailwind or headwind.
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Math Problem Analysis
Mathematical Concepts
Cost Functions
Rational Expressions
Optimization
Formulas
C(x) = 150 + x/15 + 34,000/x
Theorems
Optimization through derivatives
Critical points of a function
Suitable Grade Level
Grade 10-12 (Calculus or Advanced Algebra)
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