Airplane Distance

[nycmathdad]

An airplane flies from
Naples, Italy, in a
straight line to Rome,
Italy, which is
120 kilometers north
and 150 kilometers
west of Naples. How
far does the plane fly?

I am thinking the distance formula applies here.
The points are (0, 120) and (150, 0).

D = plane distance



Is this the correct set up?

 

[Harry_the_cat]

Just draw N and P as vertices of a right triangle and use Pythagoras.

What do the points you've mentioned represent?

 

[nycmathdad]

Just draw N and P as vertices of a right triangle and use Pythagoras.

What do the points you've mentioned represent?

I honestly just guessed the points.

The set up is (120)^2 + (150)^2 = P^2, where P is the distance the plane has traveled.

 

[Harry_the_cat]

There should be no guesswork involved. Have you drawn a diagram?

 

[JeffM]

Your work is correct, but you should know why. Otherwise, you will not know when you can use this method.

Old-fashioned (known for last 2500 years) . North and west are at right angles. Distance west is 150 km; distance north is 120 km. Therefore, by Euclidean geometry, distance

[MATH]\sqrt{150^2 + 120^2} \approx 192.[/MATH]
Solving problems by geometry is hard. So new-fashioned method (known for less than 400 years). We use a Cartesian coordinate system. We pick an arbitrary but hopefully convenient origin (you chose Naples, but you could just as conveniently have chosen Rome). We use the distance formula

[MATH]\sqrt{(150 - 0)^2 + (120 - 0)^2} \approx 192.[/MATH]
In this very simple problem, the old-fashioned method is less work. But for problems that are only slightly harder, the old-fashioned method (using Euclid) is much harder.

In either case, you should sketch a diagram to see what you are doing.

 

[nycmathdad]

There should be no guesswork involved. Have you drawn a diagram?

I have not drawn a diagram but I will when time allows.

 

[nycmathdad]

Your work is correct, but you should know why. Otherwise, you will not know when you can use this method.

Old-fashioned (known for last 2500 years) . North and west are at right angles. Distance west is 150 km; distance north is 120 km. Therefore, by Euclidean geometry, distance

[MATH]\sqrt{150^2 + 120^2} \approx 192.[/MATH]
Solving problems by geometry is hard. So new-fashioned method (known for less than 400 years). We use a Cartesian coordinate system. We pick an arbitrary but hopefully convenient origin (you chose Naples, but you could just as conveniently have chosen Rome). We use the distance formula

[MATH]\sqrt{(150 - 0)^2 + (120 - 0)^2} \approx 192.[/MATH]
In this very simple problem, the old-fashioned method is less work. But for problems that are only slightly harder, the old-fashioned method (using Euclid) is much harder.

In either case, you should sketch a diagram to see what you are doing.

I truly appreciate all your help and math notes. I have to study thanks to you.