A plane crosses the Atlantic Ocean (3000 miles) with an airspeed of 500 miles per hour. The cost C (in dollars) per
passenger is given by
C(x) = 100 + x/9 +36,000/x
where x is the ground speed (airspeed + wind).
(a) What is the cost when the ground speed is 370 miles per hour, 520 miles per hour?
(b) Find the domain of C.
(c) Use a graphing calculator to graph the function C=C(x).
(d) Create a TABLE with Tbl Start = 0 and ATbl = 50.
(e) To the nearest 50 miles per hour, what ground speed minimizes the cost per passenger?

The cost per passenger at 370 mph and 520 mph are $238.41 and $227.01 respectively.
The domain of C is x > 0
The ground speed that minimizes cost is 569 mph

Given

[tex]C(x) = 100 + \frac x9 + \frac{36000}{x}[/tex]

(a) The ground speed at 370 mph and 520 mph

This means that: [tex]x = 370[/tex] and [tex]x = 520[/tex]

When [tex]x= 370[/tex], we have:

[tex]C(370) = 100 + \frac{370}9 + \frac{36000}{370}[/tex]

[tex]C(370) = 238.41[/tex]

When [tex]x = 520[/tex], we have:

[tex]C(520) = 100 + \frac{520}9 + \frac{36000}{520}[/tex]

[tex]C(520) = 227.01[/tex]

(b) The domain:

We have:

[tex]C(x) = 100 + \frac x9 + \frac{36000}{x}[/tex]

For the plane to move, the ground speed (x) must be greater than 0.

So, the domain of C is: [tex]x > 0[/tex]

(c) See attachment for the graph of c(x)

(d) A table of x = 0 and x = 50.

When [tex]x = 0[/tex], we have:

[tex]C(x) = 100 + \frac x9 + \frac{36000}{x}[/tex]

[tex]C(0) = 100 + \frac 09 + \frac{36000}{0}[/tex]

[tex]C(0) = und efine d[/tex]

When [tex]x = 50[/tex], we have:

[tex]C(x) = 100 + \frac x9 + \frac{36000}{x}[/tex]

[tex]C(50) = 100 + \frac{50}9 + \frac{36000}{50}[/tex]

[tex]C(50) = 825.56[/tex]

So, the table is:

[tex]\left[\begin{array}{cc}x&C(x)\\0& und e fin e d\\50&825.56\end{array}\right][/tex]

(e) The speed that minimize cost

[tex]C(x) = 100 + \frac x9 + \frac{36000}{x}[/tex]

Differentiate

[tex]C'(x) = \frac{1}{9} - \frac{36000}{x^2}[/tex]

Equate to 0

[tex]\frac{1}{9} - \frac{36000}{x^2} = 0[/tex]

Collect like terms

[tex]\frac{36000}{x^2} = \frac{1}{9}[/tex]

Cross multiply

[tex]x^2 = 36000 \times 9[/tex]

[tex]x^2 = 324000[/tex]

Take square roots

[tex]x = 569.20997883[/tex]

Approximate to the nearest 50

[tex]x = 569[/tex]