SOLUTION: A Boeing 747 crosses the Atlantic Ocean (3000 miles) with an airspeed of 500 miles per hour. The cost C (in dollars) per passenger is given by the formula C(x) = 100 + (x/100

Question 1208041: A Boeing 747 crosses the Atlantic Ocean (3000 miles) with an airspeed of 500 miles per hour. The cost C (in dollars) per passenger is given by the formula

C(x) = 100 + (x/100) + (36,000/x), where x is the ground speed (airspeed x wind and - wind).

To the nearest 50 Mike's per hour, what ground speed minimizes the cost per passenger?

The question is asking to find x.

What is the set up?
Answer by math_tutor2020(3816) (Show Source): You can put this solution on YOUR website!

Answer: 1900 mph

Explanation

Apply the derivative with respect to x to go from
C(x) = 100 + (x/100) + (36000/x)
to
C ' (x) = 0.01 - 36000x^(-2)
I skipped steps and will leave the scratch work for the student to do.
This solution is basically an outline of sorts.

Set C ' (x) equal to zero and solve for x.
We do this because the local min occurs when the tangent line is 0 (i.e. when the tangent is horizontal).
I'll let the student do the 1st derivative test or 2nd derivative test to confirm that a local min exists on the interval x > 0.

If you were to solve
0.01 - 36000x^(-2) = 0
then you should get roughly
x = 1897.366596 when considering x > 0 only.
I have confirmed this with GeoGebra. Desmos is another good choice.
If you are familiar with a TI83 or similar, then you can use that as well. There are many options to pick from.

The last step is to round 1897.366596 to the nearest multiple of 50 to get the answer 1900 mph.
Unfortunately I don't know much about aviation to know if 1900 mph is a realistic ground speed or not. It seems a bit large.

Side note: for students not familiar with calculus just yet, you'll need to rely solely on graphing calculator technology.

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