Q. 33 - Precalculus Enhanced with Graphing Utilities

Question

Cost of Trans-Atlantic Travel 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 -

$C\left(x\right)=100+\frac{x}{10}+\frac{36000}{x}$

Where x is the ground speed $(airspeed + wind).$

(a). Use a graphing utility to graph the function C = C(x).

(b). Create a TABLE with TblStart = 0 and $\Delta Tbl=50.$

(c). To the nearest 50 miles per hour, what ground speed minimizes the cost per passenger?

Step-by-Step Solution

Answer

(a). 

(b).  

x	$C\left(x\right)=100+\frac{x}{10}+\frac{36000}{x}$
1	36100.1
50	825.0
100	470.0
150	355.0
200	300.0
250	269.0
300	250.0
350	237.9
400	230.0
450	225.0
500	222.0
550	220.5
600	220.0
650	220.4
700	221.4
750	223.0
800	225.0
850	227.4
900	230.0
950	232.9
1000	236.0
1050	239.3
1100	242.7
1150	246.3
1200	250.0
1250	253.8
1300	257.7
1350	261.7
1400	265.7
1450	269.8
1500	274.0
1550	278.2
1600	282.5
1650	286.8
1700	291.2
1750	295.6

(c).

The ground speed minimized cost per passenger will be -  600 mile/hours

Part (a). Step 1. Given Data

Function -  $C\left(x\right)=100+\frac{x}{10}+\frac{36000}{x}$

Part (a). Step 2. To Find

 Use a graphing utility to graph the function C = C(x). 

Part (a). Step 3. Explanation

Using the graphing utility graph which is -

Part (b). Step 1. Given Data

Function -  $C\left(x\right)=100+\frac{x}{10}+\frac{36000}{x}$

Part (b). Step 2. To Find

Create a TABLE with TblStart = 0 and $\Delta Tbl=50.$

Part (b). Step 3. Explanation

x	$C\left(x\right)=100+\frac{x}{10}+\frac{36000}{x}$
1	36100.1
50	825.0
100	470.0
150	355.0
200	300.0
250	269.0
300	250.0
350	237.9
400	230.0
450	225.0
500	222.0
550	220.5
600	220.0
650	220.4
700	221.4
750	223.0
800	225.0
850	227.4
900	230.0
950	232.9
1000	236.0
1050	239.3
1100	242.7
1150	246.3
1200	250.0
1250	253.8
1300	257.7
1350	261.7
1400	265.7
1450	269.8
1500	274.0
1550	278.2
1600	282.5
1650	286.8
1700	291.2
1750	295.6

Part (c). Step 1. Given Data

Function -  $C\left(x\right)=100+\frac{x}{10}+\frac{36000}{x}$

Part (c). Step 2. To Find

To the nearest 50 miles per hour, what ground speed minimizes the cost per passenger?

Part (c). Step 3. Explanation

Using the graphing utility, and the results from the table created in above, Find when y is minimized As shown in the table, When $\mathit{x}\mathbf{=}\mathbf{600}$ and $y=200$. (which is the lowest shown)

Thus, the Answer will be -  600 mile/hours.