Expert Help with High School Geometry

Relation via derivative of two angle-based probabilities for n points on a circle.
Suppose we sample n points uniformly (and independently) on the unit circle in the sense that the probability that a point lies within some circular arc is proportional to its length. For this question, we'll choose the convention to measure angles from 0 to 1 (rather than from 0 to $2\mathrm{\pi <!--\; \pi \; -->}$ when we use radians) to make the formulae come out nicer.
Then if we are given a fixed circular arc subtending an angle $0\le <!--\; \le \; -->\mathrm{\alpha <!--\; \alpha \; -->}\le <!--\; \le \; -->\frac{1}{2}$ (the bounds chosen due to technicalities from reflex angles), we know that the probability that all n points lie in this arc is ${\mathrm{\alpha <!--\; \alpha \; -->}}^n$. It is also a known result/elementary exercise that the probability that n such points lie within some arc which subtends an angle of $\mathrm{\alpha <!--\; \alpha \; -->}$ is $n{\mathrm{\alpha <!--\; \alpha \; -->}}^{n-<!--\; -\; -->1}$.
Is there some probabilistic reason that these formulae are related by differentiation with respect to $\mathrm{\alpha <!--\; \alpha \; -->}$?

Let $n>4$.
In how many ways can we choose $4$ vertices of a convex n-gon so as to form a convex quadrilateral, such that at least $2$ sides of the quadrilateral are sides of the $n$-gon?
Explain your answer, which should be expressed in terms of $n$.

Is it possible to derive Peirce's Law:
$⊢\ast [(\mathrm{\alpha <!--\; \alpha \; -->}\to \mathrm{\beta <!--\; \beta \; -->})\to \mathrm{\alpha <!--\; \alpha \; -->}]\to \mathrm{\alpha <!--\; \alpha \; -->}$
in a calculus that has modus ponens, the Deduction Theorem, Cut rule, Inconsistency effect and Principle of Indirect Proof?

Finding Volume Using Triple Integration
Set up and evaluate a triple integral to find the volume of the region bounded by the paraboloid $z=1-<!--\; -\; -->\frac{x^2}{9}-<!--\; -\; -->\frac{y^2}{100}$ and the xy-plane.
I understand I'm finding the volume of a paraboloid that forms a "dome" over the xy-plane. Moreover, I can see the paraboloid intersects with the xy-plane to form an ellipse given by $\frac{x^2}{9}-<!--\; -\; -->\frac{y^2}{100}=1$.
I have tried setting this up using rectangular coordinates but the integral started looking extremely messy. I then tried spherical coordinates but had trouble find the upper bound of $\mathrm{\rho <!--\; \rho \; -->}$. Specifically, I can't seem to successfully translate the rectangular equation $z=1-<!--\; -\; -->\frac{x^2}{9}-<!--\; -\; -->\frac{y^2}{100}$ to a spherical equation and isolate $\mathrm{\rho <!--\; \rho \; -->}$.

Distance between (3,4,1) and (3,6,2)?

Distance between (-6,-6) and (-5,13)?

Finding the volume of an elliptical cone with double integrals
For my class I need to find the volume of an elliptical cone bounded by $z=\sqrt{9x^2+y^2}$ and the plane $z=2$. My thought process was to integrate the equation for $z=\sqrt{9x^2+y^2}$ over the region bounded by the projection onto the xy-plane. This is the integral I set up:
${\int <!--\; \int \; -->}_{-<!--\; -\; -->2}^2{\int <!--\; \int \; -->}_{-<!--\; -\; -->\frac{1}{3}\sqrt{4-<!--\; -\; -->y^2}}^{\frac{1}{3}\sqrt{4-<!--\; -\; -->y^2}}\sqrt{9x^2+y^2}dxdy$
To check my answer, I looked up and found that the volume of an elliptical cone can be found using the equation:
$V=\frac{1}{3}\mathrm{\pi <!--\; \pi \; -->}abh$
When I checked my answer I got from the double integral, I found that it is 4 times what it should be. Can anyone explain to me what I did wrong?

Could someone explain to me why the following holds?
$\frac{\text{Area of sector}}{\text{Area of circle}}=\frac{\mathrm{\theta <!--\; \theta \; -->}}{2\mathrm{\pi <!--\; \pi \; -->}}$
When deriving the area of a sector my book just quotes the above but doesn't explain why it holds. Could someone explain?

Also, a similar argument is used when deriving arc length, which I don't understand.

By the way, I know that $\mathrm{\theta <!--\; \theta \; -->}$ is the angle subtended by the arc and $2\mathrm{\pi <!--\; \pi \; -->}$ is the angle in a full circle.

How does the ratio of the areas make it equivalent to the ratio of their angles?

Suppose a certain exam is classified as either difficult (with probability 90/92) or fair (with probability 2/92). Exams are taken one after the other. What is the probability that at least 4 difficult exams will occur before the first fair one?

What is the distance between (4.9, 2.9) and (-3.0, 4.9)?

Probability of Binomial twice of Geometric
I've come up with an interesting result:
Let X be the amount of failures of Bernoulli(p) until we get (p).
$X=Geo(p)$
$B=Bin(2X,\frac{1}{2})$
$P(B=X)=?$
Turns out: $P(B=X)=\sqrt{p}$
I found it using the Taylor expansion of $\frac{1}{\sqrt{1-<!--\; -\; -->p}}$, where the coefficient of $p^i$ turns out to be $P(Bin(2i,\frac{1}{2})=i)$.
I would like to see a probabilistic proof of this result.
Explanation of the process in words:
Roll a die with probability p of getting "X". Each time that we don't get "X", toss 2 balanced coins and accumulate the number of heads and tails. When you get "X", check if you got the same amount of heads and tails.
Programmatic explanation:
$i=0$
$while(!bernuli(p))\text{i++;}$
$Bin(2i,\frac{1}{2})\stackrel{?}{=}i$
Example:
If we succeed immediately (with probability p), $X=0$, and $P(Bin(2\cdot <!--\; \cdot \; -->0,\frac{1}{2})=0)=1$, thus $P(B=0)=1$, thus contributing p to the conditional sum, $p<\sqrt{p}$, and everything is alright.

Proof
$5\mid <!--\; \mid \; -->n⟺<!--\; ⟺\; -->5\mid <!--\; \mid \; -->n^2$

Finding the volume of a region below a plane and above a paraboloid (Triple integrals)
I have to find the volume below the plane $z=3-<!--\; -\; -->2y$ and above the paraboloid $z=x^2+y^2$.
Integrating by z first, it looks like the "arrow" I draw parallel to z-axis enters the region at $z=x^2+y^2$ and exits the region at $z=3-<!--\; -\; -->2y$. So ${\int <!--\; \int \; -->}_{x^2+y^2}^{3-<!--\; -\; -->2y}1dz$.
How I am supposed to find the other two integrals?

What is the distance between (6,5) and (1,7)?

Distance between (-4,7) and (-5,13)?

Distance between (-7,8) and (-2,1)?

In a convex quadrilateral $ABCD$, we have points $P\in <!--\; \in \; -->AB;R\in <!--\; \in \; -->AD$. Lines $PD$ and $RB$ intersect at a point $S$. Assume we can inscribe a circle in each of the quadrilaterals $APSR$ and $CDSB$. How do I prove that we can inscribe a circle in $ABCD$

“Area metric” and “Hausdorff metric” are not equivalent on all closed polygons, but equivalent on convex closed polygons
Suppose X is the set of all closed polygons, $d_{\mathrm{\Delta <!--\; \Delta \; -->}}$ is the “area metric” defined by the area sum of the symmetric difference of two closed polygons, and dH is the Hausdorff metric on X. How should I prove that $d_{\mathrm{\Delta <!--\; \Delta \; -->}}$ and $d_H$ do not generate the same topology on X? Also why do they generate the same topology on the subset of convex polygons? I tried to visualize how a typical open ball in both metrics looks but this seems rather impossible.

Finding the volume of revolution using the method of shells
I'm trying to find the volume of the solid generated by revolving the region bounded by $y=x^2$ and $y=6x+7$ about x-axis using the shell method. I applied the method and I got 15864/5 multiplied by $\mathrm{\pi <!--\; \pi \; -->}$ but it's not correct.
Details: I integrated ${\int <!--\; \int \; -->}_1^{49}y(\sqrt{y}-<!--\; -\; -->\frac{y-<!--\; -\; -->7}{6})dy$

Distance between (2,3,5) and (2,7,4)?