Math Is Fun Forum

mathxyz

Member

From: Brooklyn, NY

Registered: 2024-02-24

Posts: 1,053

Find Ground Speed of Plane

A Boeing 747 crosses the Atlantic Ocean (3000 miles) with an airspeed of 500 mph. The cost C (in dollars) per passenger is given by

C(x) = 100 + (x/10) + (36,000)/x, where x is the ground speed (airspeed plus or minus wind).

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

I say let C(x) = 0 and solve for x, the ground speed.

You say?

Bob

Administrator

Registered: 2010-06-20

Posts: 10,619

Re: Find Ground Speed of Plane

Try the graph here:

https://www.mathsisfun.com/data/function-grapher.php

You'll need to drag around and zoom out to find the curve.  It never crosses the x axis so you won't find a solution that way.  With calculus you can find the minimum. Without it you'll have to read off from the graph.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

mathxyz

Member

From: Brooklyn, NY

Registered: 2024-02-24

Posts: 1,053

Re: Find Ground Speed of Plane

Try the graph here:

https://www.mathsisfun.com/data/function-grapher.php

You'll need to drag around and zoom out to find the curve.  It never crosses the x axis so you won't find a solution that way.  With calculus you can find the minimum. Without it you'll have to read off from the graph.

Bob

I tried to graph but couldn't find the answer to the nearest 50 mph. Can you show me the calculus way?

Last edited by mathxyz (2024-03-16 11:42:19)

Bob

Administrator

Registered: 2010-06-20

Posts: 10,619

Re: Find Ground Speed of Plane

To explain fully would take weeks as there's a lot to learn with differential calculus.  I'll give you a potted version.

If you know the equation for a function then you can draw its graph.  It's useful to know how the graph's gradient behaves. In particular, if there's a maximum or minimum you can find the x value of that point by investiagting when the gradient is zero.

One helpful thing about differentiating is you can treat the three bits of this function separately and find the gradient of each bit, then add them together.

C = 100 would be a horizontal line so its gradient is zero.

C = x/10 is a sloping line with a gradient of 1/10

The last part would take a few weeks to explain so I'll just say what it is:

So the complete gradient function is

For zero gradient

(There is a negative value too but it won't fit the problem)

Hope that helps,

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

mathxyz

Member

From: Brooklyn, NY

Registered: 2024-02-24

Posts: 1,053

Re: Find Ground Speed of Plane

To explain fully would take weeks as there's a lot to learn with differential calculus.  I'll give you a potted version.

If you know the equation for a function then you can draw its graph.  It's useful to know how the graph's gradient behaves. In particular, if there's a maximum or minimum you can find the x value of that point by investiagting when the gradient is zero.

One helpful thing about differentiating is you can treat the three bits of this function separately and find the gradient of each bit, then add them together.

C = 100 would be a horizontal line so its gradient is zero.

C = x/10 is a sloping line with a gradient of 1/10

The last part would take a few weeks to explain so I'll just say what it is:

So the complete gradient function is

For zero gradient

(There is a negative value too but it won't fit the problem)

Hope that helps,

Bob

Thank you, Bob.

KerimF

Member

From: Aleppo-Syria

Registered: 2018-08-10

Posts: 239

Re: Find Ground Speed of Plane

Try the graph here:

https://www.mathsisfun.com/data/function-grapher.php

You'll need to drag around and zoom out to find the curve.  It never crosses the x axis so you won't find a solution that way.  With calculus you can find the minimum. Without it you'll have to read off from the graph.

Bob

Hi Bob,
This 'grapher' page doesn't seem to work!
Kerim

Edited:
It didn't when I entered y as: 100+x/10+36000/x
It did when enter it as: x/10+36000/x+100

Last edited by KerimF (2024-03-17 02:39:37)

---

Every living thing has no choice but to execute its pre-programmed instructions embedded in it (known as instincts).
But only a human may have the freedom and ability to oppose his natural robotic nature.
But, by opposing it, such a human becomes no more of this world.

mathxyz

Member

From: Brooklyn, NY

Registered: 2024-02-24

Posts: 1,053

Re: Find Ground Speed of Plane

Try the graph here:

https://www.mathsisfun.com/data/function-grapher.php

You'll need to drag around and zoom out to find the curve.  It never crosses the x axis so you won't find a solution that way.  With calculus you can find the minimum. Without it you'll have to read off from the graph.

Bob

Hi Bob,
This 'grapher' page doesn't seem to work!
Kerim

Edited:
It didn't when I entered y as: 100+x/10+36000/x
It did when enter it as: x/10+36000/x+100

I noticed that, too.

Try desmos.com or mathway.com for graphing.

Last edited by mathxyz (2024-03-18 01:40:22)

amnkb

Member

Registered: 2023-09-19

Posts: 253

Re: Find Ground Speed of Plane

Try desmos.com or mathway.com for graphing.

YOu can do graphing on your phone
Go to Google Play Store
In 'search' box enter 'Graphing calculator plus 84 83'
Install app to get TI-84 emulator on your phone
TI-84 can do things like find the minimum on a graph
(and there's lots of tutorials, how-to's, etc online for Texas Instruments TI-84)

mathxyz

Member

From: Brooklyn, NY

Registered: 2024-02-24

Posts: 1,053

Re: Find Ground Speed of Plane

Try desmos.com or mathway.com for graphing.

YOu can do graphing on your phone
Go to Google Play Store
In 'search' box enter 'Graphing calculator plus 84 83'
Install app to get TI-84 emulator on your phone
TI-84 can do things like find the minimum on a graph
(and there's lots of tutorials, how-to's, etc online for Texas Instruments TI-84)

Good idea. I will try that.

KEEP IN MIND

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