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).
(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	$284.50
550	$284.20
600	$285.00

(round to two decimal places as needed.)
(e) to the nearest 50 miles per hour, what ground speed minimizes the cost per passenger?
the cost per passenger is minimized at $\boldsymbol{\square}$ miles per hour.

Explanation:

Step1: Calculate C(360)

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

Step2: Calculate C(490)

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

Step3: Find domain of C(x)

Ground speed $x>0$ (cannot be 0/negative, as division by 0 is undefined, and speed can't be negative).
Domain: $x\in(0,+\infty)$

Step4: Fill table values

Use given $C(x)$ formula for each $x$:

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

Step5: Identify minimizing speed

Compare table values: $286.25>284.50>284.20<285.00$. The smallest cost is at $x=550$.

Answer:

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

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

(e) 550 miles per hour