Question

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
$c(x)=150+\frac{x}{8}+\frac{36,000}{x}$
where x is the ground speed (airspeed ± wind).
(a) what is the cost when the ground speed is 360 miles per hour, 490 miles per hour?
(b) find the domain of c.
(c) use a graphing calculator to graph the function $c = c(x)$.
graph of c?
a.
b.
c.
d.
(d) create a table with tblstart = 0 and $\delta$tbl = 50.
(e) to the nearest 50 miles per hour, what ground speed minimizes the cost per passenger?
(d) create a table as described, with tblstart = 0 and $\delta$tbl = 50. complete the following table with the table values from your graphing calculator.

x	c(x)
500	$\square$
550	$\square$
600	$\square$

(round to two decimal places as needed.)

Explanation:

Step1: Calculate C(360)

Substitute $x=360$ into $C(x)$:
$$C(360)=150+\frac{360}{8}+\frac{36000}{360}$$
$$C(360)=150+45+100=295$$

Step2: Calculate C(490)

Substitute $x=490$ into $C(x)$:
$$C(490)=150+\frac{490}{8}+\frac{36000}{490}$$
$$C(490)=150+61.25+73.47\approx284.72$$

Step3: Find domain of C(x)

Ground speed $x>0$ (cannot be 0/negative, as it represents speed).

Step4: Identify correct graph of C(x)

The function $C(x)=150+\frac{x}{8}+\frac{36000}{x}$ is a sum of a linear increasing term and a reciprocal decreasing term. It decreases first, reaches a minimum, then increases slowly. This matches graph A.

Step5: Calculate table values

Substitute $x=450,500,550,600$ into $C(x)$:

• $C(450)=150+\frac{450}{8}+\frac{36000}{450}=150+56.25+80=286.25$
• $C(500)=150+\frac{500}{8}+\frac{36000}{500}=150+62.5+72=284.50$
• $C(550)=150+\frac{550}{8}+\frac{36000}{550}=150+68.75+65.45\approx284.20$
• $C(600)=150+\frac{600}{8}+\frac{36000}{600}=150+75+60=285.00$

Step6: Find speed minimizing cost

Use calculus or analyze the function: take derivative $C'(x)=\frac{1}{8}-\frac{36000}{x^2}$, set to 0:
$$\frac{1}{8}-\frac{36000}{x^2}=0$$
$$x^2=36000\times8=288000$$
$$x=\sqrt{288000}\approx536.66$$
Rounded to nearest 50 is 550 mph.

Answer:

(a) Cost at 360 mph: $\$295.00$; Cost at 490 mph: $\$284.72$
(b) Domain: All positive real numbers, $x>0$
(c) Correct graph: A
(d)

x	C(x)
500	$\$284.50$
550	$\$284.20$
600	$\$285.00$

(e) 550 miles per hour