TakahiroMaeda

The answer is 40

The Newton method is quite the complicated method if you don't have an advance knowledge of Maths. However, given that you have asked, I will try my best to teach you the Newton Method.

Now then let is get started. The Newton's Method uses a very repetitive method of approaching a root:

$${\sqrt{}}$$

The root that you wish to solve or locate will depend what your initial number will be. In this example I will use the most commonly used pronumeral to replace a number. The equation you are looking for is indicated below: 

$$x_n_+_1=(x_n)-f(x_n)/f'(x_n)$$

Looks complicated right?

I'll try to walk you through.

In the equation $$x_n$$ is the value of the $$x$$ value. $$f(x_n)$$ is the same $$x_n$$.

I am very bad at wording... Please excuse me.. I never exceled at Engrish.

$$f'(x_n)$$ is the slope or as known as the derivative, at

$$x_n. x_n_+_1$$

represents the next x-value that you are trying to find.

$$f'(x_n)$$ the derivative represents $$f(x)/dx (dx = delta-x)$$. Therefore, the term $$f(x)/f'(x)$$ represents a value of dx.

Right now.. I am diving into the unknown now... Hopefully, I still know what I am doing >_<

$$\frac{f(x)}{f'(x)}=\frac{f(x)}{f(x)/dx}=dx$$

...and that is all I remember from this method. Anyone could help me out? :P

Okay Lets do this:

Distance flown by the Airplane = 3500 Kilometers

Distance traveled by the train = 600 Kilometers

Lets make x the taken by the Airplane

Because of this, it would make sense that the time taken to travel 600 Kilometers is also x.

Now to find out the speed of the Airplane, the forumla is 

$${\frac{{\mathtt{3\,500}}}{{\mathtt{x}}}}$$

Now to find out the speed of the Train, the forumla is

$${\frac{{\mathtt{600}}}{{\mathtt{x}}}}$$

Now let us use what we already know.

$${\frac{{\mathtt{3\,500}}}{{\mathtt{X}}}} = {\mathtt{5}}{\mathtt{\,\times\,}}\left({\frac{{\mathtt{600}}}{{\mathtt{X}}}}\right){\mathtt{\,\small\textbf+\,}}{\mathtt{50}}$$

$${\frac{{\mathtt{3\,500}}}{{\mathtt{x}}}} = \left({\frac{{\mathtt{3\,000}}}{{\mathtt{x}}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{50}}\right)$$

Now we need to eradicate the x

So we multiply x to both sides.

$${\mathtt{3\,500}} = {\mathtt{3\,000}}{\mathtt{\,\small\textbf+\,}}{\mathtt{50}}{\mathtt{\,\times\,}}{\mathtt{X}}$$

Now we need to use some basic algebra knowledge to solve this. Let us move x to one side and the numbers on the other. We will get

$${\mathtt{x}} = {\mathtt{10}}$$

Now, lets go back to what we know already know and substitute.

For the Plane:

$${\mathtt{SPEED}} = {\frac{{\mathtt{3\,500}}}{{\mathtt{X}}}}$$

$${\mathtt{SPEED}} = {\frac{{\mathtt{3\,500}}}{{\mathtt{10}}}}$$

$${\mathtt{SPEED}} = {\mathtt{350}}{KM}$$per hour

For the Train:

$${\mathtt{SPEED}} = {\frac{{\mathtt{600}}}{{\mathtt{X}}}}$$

$${\mathtt{SPEED}} = {\frac{{\mathtt{600}}}{{\mathtt{10}}}}$$

$${\mathtt{SPEED}} = {\mathtt{60}}{KM}$$ per hour.

There we go.

The Speed of the plane was 350KM per hour 

and the speed of the train was 60 Km per hour

Nah, I understand. However, generally when they ask a question like this, they are looking for a detailed answer. So they don't ask again, they can do it themselves

I'll save you the trip to the calculator!

$${\frac{{\mathtt{1.666\: \!666\: \!66}}}{{\mathtt{3.5}}}} = {\frac{{\mathtt{10}}}{{\mathtt{21}}}} = {\mathtt{0.476\: \!190\: \!474\: \!285\: \!714\: \!3}}$$

To work out the Volume of the Pyramid, the equation is listed below.

$${\mathtt{Volume}} = {\frac{\left({\mathtt{l}}{\mathtt{\,\times\,}}{\mathtt{w}}{\mathtt{\,\times\,}}{\mathtt{h}}\right)}{{\mathtt{3}}}}$$

Right then!

Lets solve this together!

First of all, we will need to turn all of the Mixed Fractions into improper fraction for ease of understanding.

To turn an Mixed Number or Mixed Fraction into an improper fraction the steps are quite simple. First look at the denominator. Lets use an Mixed Number in the equation.

2 1/3.

The denominator is 3 correct? Now we need to look at the Whole number. That is 2. Now we just multiply then together.

$${\mathtt{2}}{\mathtt{\,\times\,}}{\mathtt{3}}$$

What does that equal? 6.

Now we just add the 6 to the numerator. So we get a fraction of:

$${\frac{{\mathtt{7}}}{{\mathtt{3}}}}$$

Now we do that for all the mixed numbers.

So we should end up with:

$${\frac{{\mathtt{7}}}{{\mathtt{3}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{8}}}{{\mathtt{5}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{19}}}{{\mathtt{6}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{11}}}{{\mathtt{4}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{22}}}{{\mathtt{5}}}}$$

Now we reach the hardest part of solving this equation.

What is the LCM of all of these numbers?

LCM is Lowest Common Multiple.

So what is the lowest number that all 5 of these numbers go into. 

Let me guide you through this step.

Lets start with any number.

20?

Okay. Does 3 Go into 20?

Sadly no. So this does not qualify as the Lowest Common Multiple.

60?

Does 3 Go into 60? Yes!

Does 5 Go into 60? Yes!

Does 4 Go into 60? Yes!

Does 6 Go into 60? Yes!

We skip the other one, because there are two 5's, making writing it again useless.

How then we found a number, we will need to calculate the amount of the number goes to the other number.

So.. How many 3's go into 60?

20.

5 to 60?

12.

4 to 60?

15.

6 to 60?

10. 

Now lets utilise this!

Back to the Equation.

$${\frac{{\mathtt{7}}}{{\mathtt{3}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{8}}}{{\mathtt{5}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{19}}}{{\mathtt{6}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{11}}}{{\mathtt{4}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{22}}}{{\mathtt{5}}}}$$

We want to the the demoninator the LCM. Or in this equation 60.

Go do that, we must multiply what we do to the demoninator also to the Numerator.

So. For Example:

$${\frac{{\mathtt{7}}}{{\mathtt{3}}}}$$

What do we need to do with the Demoninator to get it to 60?

Multiply by 20!

So we multiply both the top and the bottom with 20.

$${\frac{\left({\mathtt{7}}{\mathtt{\,\times\,}}{\mathtt{20}}\right)}{\left({\mathtt{3}}{\mathtt{\,\times\,}}{\mathtt{20}}\right)}}$$

We get:

$${\frac{{\mathtt{140}}}{{\mathtt{60}}}}$$

Now do the same to all the other fractions so we will get:

$${\frac{{\mathtt{140}}}{{\mathtt{60}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{96}}}{{\mathtt{60}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{190}}}{{\mathtt{60}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{165}}}{{\mathtt{60}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{264}}}{{\mathtt{60}}}}$$

Now add them all together.

$${\frac{{\mathtt{140}}}{{\mathtt{60}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{96}}}{{\mathtt{60}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{190}}}{{\mathtt{60}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{165}}}{{\mathtt{60}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{264}}}{{\mathtt{60}}}} = {\frac{{\mathtt{57}}}{{\mathtt{4}}}} = {\mathtt{14.25}}$$

There we are! The Answer.

14.25

You answered yourself. ^_^

In this case would it be safe to assume this:

$${{\mathtt{642.348}}}^{\left({\frac{{\mathtt{1}}}{{\mathtt{3.33}}}}\right)} = {\mathtt{6.969\: \!061\: \!341\: \!126\: \!795}}$$

Because Radical usualy means root. And Order.... Well... Order? lol?