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

$C\left(x\right)=100+\frac{x}{10}+\frac{36,000}{x}$

where x is the ground speed (airspeed$\pm$wind).

(a) What is the cost per passenger for quiescent (no wind) conditions?

(b) What is the cost per passenger with a head wind of 50 miles per hour?

(c) What is the cost per passenger with a tail wind of 100 miles per hour?

(d) What is the cost per passenger with a head wind of 100 miles per hour?

Short answer

(a) The cost per passenger for quiescent (no wind) conditions is$\$222$.

(b) The cost per passenger with a head wind of 50 miles per hour is$\$225$

(c) The cost per passenger with a head tail of 100 miles per hour is$\$220$

(d) The cost per passenger with a head wind of 100 miles per hour is$\$230$

Step-by-step solution

Step 1. Given Information

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

$C\left(x\right)=100+\frac{x}{10}+\frac{36,000}{x}$

where x is the ground speed (airspeed$\pm$wind).

(a) What is the cost per passenger for quiescent (no wind) conditions?

(b) What is the cost per passenger with a head wind of 50 miles per hour?

(c) What is the cost per passenger with a tail wind of 100 miles per hour?

(d) What is the cost per passenger with a head wind of 100 miles per hour?

Part (a) Step 1. The given function is C(x)=100+x10+36,000x.

We have to find the cost per passenger for quiescent (no wind) conditions.

We know the value of $x=(\mathrm{airspeed}\pm \mathrm{wind})$

In no wind conditions the value of wind$=0$

Part (a) Step 2. In this part the value of x=500

$$\begin{gathered} C\left(500\right)=100+\frac{500}{10}+\frac{36,000}{500} \\ C\left(500\right)=100+50+72 \\ C\left(500\right)=222 \end{gathered}$$

Part (b) Step 1. We have to find the cost per passenger with a head wind of 50 miles per hour.

The value of $x=(\mathrm{airspeed}\pm \mathrm{wind})$

We have to find a head wind of 50 miles per hour.

So the value of

$$\begin{gathered} x=500-50 \\ x=450 \end{gathered}$$

Part (b) Step 2. Now putting the value in the given function.

$$\begin{gathered} C\left(450\right)=100+\frac{450}{10}+\frac{36,000}{450} \\ C\left(450\right)=100+45+80 \\ C\left(450\right)=225 \end{gathered}$$

Part (c) Step 1. We have to find the cost per passenger with a tail wind of 100 miles per hour.

The value of$x=(\mathrm{airspeed}\pm \mathrm{wind})$

We have to find a tail wind of 100 miles per hour.

So the value of

$$\begin{gathered} x=500+100 \\ x=600 \end{gathered}$$

Part (c) Step 2. Now putting the value in the given function.

$$\begin{gathered} C\left(600\right)=100+\frac{600}{10}+\frac{36,000}{600} \\ C\left(600\right)=100+60+60 \\ C\left(600\right)=220 \end{gathered}$$

Part (d) Step 1. We have to find the cost per passenger with a head wind of 100 miles per hour.

The value of$x=(\mathrm{airspeed}\pm \mathrm{wind})$

We have to find a head wind of 100 miles per hour.

So the value of

$$\begin{gathered} x=500-100 \\ x=400 \end{gathered}$$

Part (d) Step 2. Now putting the value in the given function.

$$\begin{gathered} C\left(400\right)=100+\frac{400}{10}+\frac{36,000}{400} \\ C\left(400\right)=100+40+90 \\ C\left(400\right)=230 \end{gathered}$$