An airplane crosses the Atlantic Ocean​ (3000 miles) with an airspeed of 600 miles per hour. The cost C​ (in dollars) per passenger is given by

C(x)=150+x/20+36,000/x

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

What is the cost per passenger with a head wind of 50 miles per​ hour?

What is the cost per passenger with a tail wind of 100 miles per​ hour?

What is the cost per passenger with a head wind of 100 miles per​ hour?

Why was the airplane always so calm?

Because it had quiescent (no wind) conditions!

For quiescent conditions, we can substitute the airspeed into the cost function:

C(x) = 150 + x/20 + 36,000/x

Substituting the airspeed of 600 miles per hour:

C(600) = 150 + 600/20 + 36,000/600

Simplifying the equation, we get:

C(600) = 150 + 30 + 60

C(600) = 240 dollars per passenger in quiescent conditions.

Now let's tackle the windy situations!

With a headwind of 50 miles per hour, we need to adjust the airspeed:

Adjusted airspeed = airspeed - headwind
Adjusted airspeed = 600 - 50
Adjusted airspeed = 550 miles per hour

Substituting the adjusted airspeed into the cost function:

C(550) = 150 + 550/20 + 36,000/550

Simplifying the equation, we get:

C(550) = 150 + 27.5 + 65.45

C(550) = 242.95 dollars per passenger with a headwind of 50 miles per hour.

Next up, a tailwind of 100 miles per hour!

Adjusted airspeed = airspeed + tailwind
Adjusted airspeed = 600 + 100
Adjusted airspeed = 700 miles per hour

Substituting the adjusted airspeed into the cost function:

C(700) = 150 + 700/20 + 36,000/700

Simplifying the equation, we get:

C(700) = 150 + 35 + 51.43

C(700) = 236.43 dollars per passenger with a tailwind of 100 miles per hour.

Lastly, another headwind but this time at 100 miles per hour!

Adjusted airspeed = airspeed - headwind
Adjusted airspeed = 600 - 100
Adjusted airspeed = 500 miles per hour

Substituting the adjusted airspeed into the cost function:

C(500) = 150 + 500/20 + 36,000/500

Simplifying the equation, we get:

C(500) = 150 + 25 + 72

C(500) = 247 dollars per passenger with a headwind of 100 miles per hour.

Hope that helps, and remember, even airplanes need some wind therapy now and then!

To find the cost per passenger for quiescent (no wind) conditions, we need to substitute x = 0 into the equation C(x) = 150 + x/20 + 36,000/x.

Substituting x = 0 gives us:
C(0) = 150 + 0/20 + 36,000/0.

However, division by 0 is undefined, so we cannot find the cost per passenger for quiescent conditions.

To find the cost per passenger with a headwind of 50 miles per hour, we need to substitute x = 600 - 50 = 550 into the equation C(x) = 150 + x/20 + 36,000/x.

Substituting x = 550 gives us:
C(550) = 150 + 550/20 + 36,000/550.

Simplifying the equation gives us:
C(550) = 150 + 27.5 + 65.45.

Therefore, the cost per passenger with a headwind of 50 miles per hour is:
C(550) = $242.95 per passenger.

To find the cost per passenger with a tailwind of 100 miles per hour, we need to substitute x = 600 + 100 = 700 into the equation C(x) = 150 + x/20 + 36,000/x.

Substituting x = 700 gives us:
C(700) = 150 + 700/20 + 36,000/700.

Simplifying the equation gives us:
C(700) = 150 + 35 + 51.43.

Therefore, the cost per passenger with a tailwind of 100 miles per hour is:
C(700) = $236.43 per passenger.

To find the cost per passenger with a headwind of 100 miles per hour, we need to substitute x = 600 - 100 = 500 into the equation C(x) = 150 + x/20 + 36,000/x.

Substituting x = 500 gives us:
C(500) = 150 + 500/20 + 36,000/500.

Simplifying the equation gives us:
C(500) = 150 + 25 + 72.

Therefore, the cost per passenger with a headwind of 100 miles per hour is:
C(500) = $247 per passenger.

To determine the cost per passenger for different wind conditions, we need to substitute the given airspeed values into the cost function C(x) and evaluate the expression.

1. Cost per passenger for quiescent (no wind) condition:
For a quiescent condition, there is no wind affecting the plane's airspeed. Therefore, we substitute x = 600 into the cost function C(x).
C(x) = 150 + x/20 + 36,000/x
C(600) = 150 + 600/20 + 36,000/600
C(600) = 150 + 30 + 60
C(600) = 240 dollars

The cost per passenger for quiescent conditions is 240 dollars.

2. Cost per passenger with a headwind of 50 miles per hour:
With a headwind, the airspeed is reduced by the speed of the wind. To find the effective airspeed, we subtract the headwind speed from the airplane's airspeed. Therefore, x = 600 - 50 = 550.
C(x) = 150 + x/20 + 36,000/x
C(550) = 150 + 550/20 + 36,000/550
C(550) = 150 + 27.5 + 65.45
C(550) = 242.95 dollars

The cost per passenger with a headwind of 50 miles per hour is 242.95 dollars.

3. Cost per passenger with a tailwind of 100 miles per hour:
With a tailwind, the airspeed is increased by the speed of the wind. To find the effective airspeed, we add the tailwind speed to the airplane's airspeed. Therefore, x = 600 + 100 = 700.
C(x) = 150 + x/20 + 36,000/x
C(700) = 150 + 700/20 + 36,000/700
C(700) = 150 + 35 + 51.43
C(700) = 236.43 dollars

The cost per passenger with a tailwind of 100 miles per hour is 236.43 dollars.

4. Cost per passenger with a headwind of 100 miles per hour:
With a headwind, the airspeed is reduced by the speed of the wind. To find the effective airspeed, we subtract the headwind speed from the airplane's airspeed. Therefore, x = 600 - 100 = 500.
C(x) = 150 + x/20 + 36,000/x
C(500) = 150 + 500/20 + 36,000/500
C(500) = 150 + 25 + 72
C(500) = 247 dollars

The cost per passenger with a headwind of 100 miles per hour is 247 dollars.