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swastigarg

@ swastigarg

joined in may 2021

3.9k

  • earth sciences
  • science

if we can't tunnel through the earth, how do we know what's at its center ?

first, scientists are pretty sure the earth's core is about 80 percent iron. how they know that is a series of educated guesses.

for starters, they can be reasonably sure of the planet's mass based on its gravitational pull. the material on the surface isn't dense enough to match up with that mass, so the rest of the earth has to be much denser.

iron, meanwhile, is one of the most prevalent elements in the universe, but it isn't all that evident in the earth's crust. since scientists would expect more iron to be in our planet and it's a fairly dense element, that leads to the conclusion that the earth's core is mostly iron. they theorize that it was pulled to the core over millions of years. by examining different seismic waves, they know that the inner part of the core is solid and the outer core is molten.

it's a lot more scientific than i've made it sound, but hopefully that gets at the gist of it.

  • math and arithmetic
  • trigonometry

how do you solve cosx - xsinx equals 0 ?

to solve this, you need to find values of x where cos(x)

=

xsin(x).



first of all, 0 is not a solution because cos(0) =

1, and sin(0) =

0. since 0 is not a solution, divide both sides of the equation by sin(x)

to get cot(x)

=

x (remember that cos divided by sin is the same as cot). the new question to answer is, when is cot(x)

=

x? using wolfram alpha, the results are

x â±9.52933440536196...
x â±6.43729817917195...
x â±3.42561845948173...
x â±0.860333589019380... there will be an infinite number of solutions.

if you'd like to do the calculation yourself (not asking wolframalpha)

then there's a trick which almost always works, even for equations which cant be done analytically.

starting with the basic equation, cos(x)

=

x*sin(x),

transpose it to a form starting with "x =".

in this case you could get: x =

1/tan(x), x =

cot(x)

or from tan(x)

=

1/x you get x =

arctan(1/x).

because i like to do my calcs

on an old calculator which only has arctan

and not arccotan

(inverse cotangent(x))

i use the last above - x =

arctan(1/x)

starting with a value like 0.5, hit the 1/x key then shift tan keys. just keep repeating those two operations and the display will converge on 0.860333. too easy. this example of the method is not a good one as it takes about 25 iterations to converge to within 0.0000001 of the right answer. it is unusually slow.

and finally, this method has only 50% chance of working first try. we were lucky picking x =

arctan(1/x). x =

1/tan(x) diverges ind the iterations do not converge on the answer.

so if you try this method on another problem and it diverges, just transpose the equation again and have another go.

starting with x^2 + x - 3 =

0,

and iterating x =

3-x^2, you find it diverges, so

try x =

sqr(3-x) which (with care and about 25 iterations) converges on 1.302775638.