merry sequoia
B→ 21. A rectangular sports field with an area of 810 m² needs to be fenced on the north and south sides with a wooden fence, and on the east and west sides with a wire mesh (see diagram). Installing one running meter of wooden fence costs 50 EUR, and wire mesh - 20 EUR. What should be the dimensions of this sports field so that the installation of the wooden fence and wire mesh costs the least?
What I understand is that x * by y should be 810
Price would be 100x + 40y
honest bluffBOT
trail forge
its a common optimization problem. the thing is, you have 2 variables: x and y
but you want to have only one variable
merry sequoia
we could get for example y = 810/x
merry sequoia
But
trail forge
and what now?
merry sequoia
amm how to get Y
trail forge
no. if you plug in some x, what does the term tell you?
merry sequoia
idk 😭
trail forge
whats x again?
merry sequoia
one side
trail forge
ah alright
trail forge
no. x is just one side, thats true
trail forge
now think about this: if we plug in x=10 for 100x + 32400/x, then what do we get?
merry sequoia
1324
trail forge
what does this number mean?
trail forge
well what do you think/guess?
merry sequoia
money maybe
trail forge
exactly
merry sequoia
for side
trail forge
the price for the whole fencing
merry sequoia
ah
trail forge
so if x=10, then it costs 1324, if we plug in x=15, we get something different and so on...
merry sequoia
Yep
trail forge
we an say that this a function. we can plug in whole lot of different x and the term will tell us how much it costs
so lets write it as a function f(x) = 100x + 32400/x
merry sequoia
Ok
trail forge
what do we want to know in the end?
merry sequoia
field dimensions that make it as cheap as possible
trail forge
ok perfect. and what does as cheap as possible in terms of this function mean?
trail forge
thats automatically in the function because of the 810/x*40
what do we want to know from the funtion?
merry sequoia
dimensions ?
trail forge
sorry for being so suspenseful
what does as cheap as possible mean if we look at the function?
merry sequoia
It's ok I just don't get it + math english isn't something I know
merry sequoia
what does *as cheap as possible* mean if we look at the function?
least amount of money
merry sequoia
minimum
trail forge
YES
merry sequoia
then derivative
trail forge
you want to know where the minimum of the function is
you want to know which x to plug in, such that the function spits out the smallest possible value
merry sequoia
so (100x^2 + 32400) / x
trail forge
then derivative
good
merry sequoia
And (100x^2 + 32400) / x = 0
trail forge
so (100x^2 + 32400) / x
thats not right
trail forge
thats the derivative, yes
merry sequoia
merry sequoia
now ammmm
trail forge
i'll give you some time
trail forge
yes, since there are no negative lengths in reallife, you can ignore the -18
so are you done now?
trail forge
how do we find that
merry sequoia
wire mesh
amm wait a sec
oh
so we got x
and since y = 810/x
y = 45
Done :DDD
Thank you very much
trail forge
yes exactly. you use the equation you began with: xy=810
merry sequoia
Actually understand something now
