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use the focus directrix form of the eqn of a parabola:
x² = 4py (or y² = 4px for the horizontal case); note that p = the distance from the vertex to the focus = the distance from the vertex to the directrix
now, mathsyperson has already done most of the work!
take x² = 16y = 4(4y), so we see p = 4.
this means our parabola (with vertex (0,0) ) has focus (0, 4) and directrix y = -4
sounds like the answer's supposed to be:
A = pi (y+5)²/4 = (pi/4) (y² + 10y + 25)
2sinxcosx + cos^2x = 1
=>
2sinxcosx - sin^2x = 0 (bc of the pythagorean identity)
=>
(2cosx - sinx)sinx = 0
=>
2cosx=sinx or sinx = 0
=>
tan x = 2 or sin x = 0
so x = arctan 2 or x = n(pi), where n is any integer.
ahgua, do not worry about dividing by cos x in the (correct) sol'n given by wcy; sin x/cos x is defined as tan x, so we do not need to worry about division by zero, the dfn of tan x already forbids it!
another way to show x=0 is a sol'n was already given by wcy in that soln; the sin x = 0 half of the soln...
right, I missed that ambiguity!
correct, mikau: it doesn't matter which comes first, not even the exponent rule... the only ordering you have to obey is PEMDAS, as MathsIsFun said.
darn... I was trying to be too fancy.... replace all the question marks in that above post with the symbol 'infinity' (or '+infinity', if you prefer)
First, recall the theorem ( a(n) -> 0 ) => ( (a(n))^(1/n) -> 0 ) , where n -> ∞ in both limits. (the proof of this is pretty straightforward, but I can do it if you like)
Then note the well known fact that (x^n)/(n!) -> 0 as n -> ∞; put this together with the above theorem and get:
x/((n!)^(1/n)) -> 0 as n -> ∞.
Since x is constant in the limit, we must have:
(n!)^(1/n) -> ∞ as n -> ∞, and it's positive infinity since n! > 1 for all n in N.
[replaced ? with ∞ for you - mathsisfun]
if you use them correctly, there is no preferred order; you should be able to work any equation using any approach (i.e. order) you wish. just make sure that you always follow the order of operation rules for arithmetic!
if you would post a specific question where you ran into trouble, I would be more than happy to explain what I'm talking about using your situation as an example...
oh and btw the second one should be: (log m) - (log n) = log (m/n)
another thing.... does 'x' in the eq'n signify multiplication? or is it a variable? you have to understand what kylekatarn is trying to say in his post: it is virtually impossible to solve one eq'n in 5 unknowns! you have to tell us which is known, and which is unknown (like wcy said, we have to know whether or not that 'e' you've used is the natural exponential!); i.e., do we solve for a? or some other variable?
I'm getting to think this isn't a serious topic... just like kylekatarn said! especially since you didn't simplify the (8+18-20) part of it before you posted....
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