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## You must cut a wire of $36cm$ to form a triangle and a rectangle in a specific place to find the minimum area.

Well, I have an wire of $36cm$ and I need to cut it in two parts, one to form an equilateral triangle, and the other to form a rectangle such that its width is two times the height. Where do I need...

mathematics

## Optimisation problem - circle and square

A piece of wire of length $20$cm is cut into $2$ parts. the first part is bent into a circle of radius $r$ in cm, the second into a square of side length $s$ in cm. a) write down an expression for...

mathematics

## Find the value of $m$ and $n$ Calculus related

Good day everyone. I really need help in this question, it kept me thinking for a few tens of minutes. The gradient of the curve $y=mx^2+\frac{n}{x}$ at point $(2,12)$ is 6. Find the value of $m$ ...

mathematics

## When does a Möbius transformation map $\Im(z)>0$ to itself?

Show Möbius transformation which maps $\Im(z)&gt;0$ to itself iff $$f(z)= \frac{az+b}{cz+d}\,,\,\,ad-bc&gt;0$$ and $a,b,c,d$ are real.

mathematics

## $\frac{(2n)!}{4^n n!^2} = \frac{(2n-1)!!}{(2n)!!}=\prod_{k=1}^{n}\bigl(1-\frac{1}{2k}\bigr)$

i cant see why we have : $$\frac{(2n)!}{4^n n!^2} = \frac{(2n-1)!!}{(2n)!!}$$ $$\dfrac{(2n-1)!!}{(2n)!!} =\prod_{k=1}^{n}\left(1-\dfrac{1}{2k}\right),$$ Even i see the notion of D...

mathematics

## Optimise problem. Find minimum amount of wire to stabilise a three leg stool

So there is a stool, the legs are apart from each other in a triangle fashion (isosceles triangle). The length between the points is 5 , 5 and 6. We want to stabilise this shaky stool by putting w...

mathematics

## Calculus / find the value of $x$ so that $f ''(x)=0$

Let $f(x)= 10xe^x$ $(a)$ Find the exact value of $x$ so that $f ''(x) = 0$. I tried: \begin{align}f'(x)&amp; = 10e^x\\f''(x)&amp;= e^x\end{align} but at that point

mathematics

## Why are alkali basalts found at rifting centres and on top of 'plumes', whereas MORBs are generally tholeiitic?

So I was essentially wondering why it was that the basalts on top of 'plumes' at 'hotspots' (sometimes called OIBs, ocean island basalts) and at continental rifting centres are enriche

earth-science

mathematics

## Existence of a random variable satisfying a condition on its distribution

Let $X, Y : [0,1] \to \mathcal{X}$ be two random variables. Here, $[0,1]$ is the interval with the Lebesgue $\sigma$-algebra and $\mathcal{X}$ is a topological space with the Borel $\sigma$-algebra...

mathematics

## How can I show that the "binary digit maps" $b_i : [0,1) \to \{0,1\}$ are i.i.d. Bernoulli random variables?

In this post What is the Lebesgue measure of the set of numbers in $[0,1]$ that has two thirds of ones in their infinite base-2 expansion? we needed the fact that if we let $b_i (x) \in \{0,1\}$ fo...

mathematics

## Identification of legitimate distribution function

I'm completely new in this, so help me please. $F(t)$ is the distribution function. $G(t)$ is the limit of $F^n(t)$ as $n\to\infty$. Is it true that $G(t)$ is the distribution function? $F(t)$ i

mathematics

## implications of unimodality of distribution

Suppose $X$ is a continuous r.v. such that $X\ge0$ and its distribution is unimodal. What sort of consequences does that entail for $X$ in terms of its other properties (e.g., moments, etc.)?

mathematics