Expert Help with High School Geometry

Distance between (6,8,2) and (3,4,1)?

We have $AB=BC$, $AC=CD$, $\mathrm{\angle <!--\; \angle \; -->}ACD={90}^{\circ <!--\; \circ \; -->}$. If the radius of the circle is 'r' , find BC in terms of r .

Finding volume of solid in one quadrant - divide total volume by 4? 8? 2?
I want to find the volume of the solid produced by revolving the region enclosed by $y=4x$ and $y=x^3$ in the first quadrant. The wording about the first quadrant confuses me but here's my work so far:
I know the volume unrestrained by quadrant is:
$V={\int <!--\; \int \; -->}_a^b\mathrm{\pi <!--\; \pi \; -->}(f(x{)}^2-<!--\; -\; -->g(x{)}^2)dx$
Where $f(x)=4x$ and $g(x)=x^3$. To find a and b, I look for the largest and smallest intersection points between the two functions:
$\begin{array}{rl}f(x)& =g(x)\\ 4x& =x^3\\ 0& =x^3-<!--\; -\; -->4x\\ & =x(x-<!--\; -\; -->2)(x+2)\\ x& \in <!--\; \in \; -->\{-<!--\; -\; -->2,0,2\}\end{array}$
Plugging all of these into the volume equation above:
$\begin{array}{rl}V& ={\int <!--\; \int \; -->}_a^b\mathrm{\pi <!--\; \pi \; -->}(f(x{)}^2-<!--\; -\; -->g(x{)}^2)dx\\ & ={\int <!--\; \int \; -->}_{-<!--\; -\; -->2}^2\mathrm{\pi <!--\; \pi \; -->}((4x{)}^2-<!--\; -\; -->(x^3{)}^2)dx\\ & =\mathrm{\pi <!--\; \pi \; -->}{\int <!--\; \int \; -->}_{-<!--\; -\; -->2}^2(16x^2-<!--\; -\; -->x^6)dx\\ & =\mathrm{\pi <!--\; \pi \; -->}({\int <!--\; \int \; -->}_{-<!--\; -\; -->2}^{2}16x^{2}dx-<!--\; -\; -->{\int <!--\; \int \; -->}_{-<!--\; -\; -->2}^2{x}^{6}dx)\\ & =\mathrm{\pi <!--\; \pi \; -->}({\left[\frac{16x^3}{3}\right]}_{-<!--\; -\; -->2}^2-<!--\; -\; -->{\left[\frac{x^7}{7}\right]}_{-<!--\; -\; -->2}^2)\\ & =\mathrm{\pi <!--\; \pi \; -->}(\frac{16(2{)}^3}{3}-<!--\; -\; -->\frac{16(-<!--\; -\; -->2{)}^3}{3}-<!--\; -\; -->\frac{(2{)}^7}{7}-<!--\; -\; -->\frac{(-<!--\; -\; -->2{)}^7}{7})\\ & =\mathrm{\pi <!--\; \pi \; -->}(\frac{128}{3}+\frac{128}{3}-<!--\; -\; -->\frac{128}{7}+\frac{128}{7})\\ & =\mathrm{\pi <!--\; \pi \; -->}(\frac{256}{3}-<!--\; -\; -->\frac{256}{7})\\ & =\mathrm{\pi <!--\; \pi \; -->}\left(\frac{1024}{21}\right)\\ & =\frac{1024\mathrm{\pi <!--\; \pi \; -->}}{21}\end{array}$
This is the volume for the entire function. I make an assumption that since I only want one quadrant and the function is symmetric about both the x- and y-axes, I simply divide it by four.
$\begin{array}{rl}{V}_{\text{whole}}& =\frac{1024\mathrm{\pi <!--\; \pi \; -->}}{21}\\ V_{\text{one quadrant}}& =\frac{1024\mathrm{\pi <!--\; \pi \; -->}}{21}\times <!--\; \times \; -->\frac{1}{4}\\ & =\frac{256\mathrm{\pi <!--\; \pi \; -->}}{21}\end{array}$
I have no way of verifying my results. Can my assumption be made, or there's a differing method I should be using here?
If I'm now working in 3D space, would I instead divide it by eight? But if I'm revolving around $x=0$, wouldn't the solid of revolution take four quadrants in 3D space, thus I should divide the total volume by 2?

What is the distance between (8,5) and (6,2)?

Distance between (3,5,-2) and (-8,5,4)?

The maximum area of a hexagon that can inscribed in an ellipse
I have to find the maximum area of a hexagon that can inscribed in an ellipse. The ellipse has been given as $\frac{x^2}{16}+\frac{y^2}{9}=1$.
I considered the circle with the major axis of the ellipse as the diameter of the circle with centre as the origin. By considering lines of $y=\pm <!--\; \pm \; -->\sqrt{3}x$, you can find the points on the circle which are vertices of a regular hexagon(including the end points of the diameter). I brought these points on the given ellipse such that the x coordinate of the vertices remain the same and only the y coordinate changes depending upon the equation of the ellipse. But that did not lead to a hexagon.
Since the question nowhere mentions of the hexagon being regular, I think it might be a irregular hexagon but any such hexagon can be drawn which encompasses most of the area of the ellipse.

Distance between (-4,0,2) and (0,4,-2)?

Probability X is odd in a geometric distribution
Let X be a Geometric distribution with parameter $p=\frac{1}{10}$.

What is the distance between (6,12) and (-6,13)?

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

Theorems that we can prove only by contradiction

Gauss-divergence theorem for volume integral of a gradient field
I need to make sure that the derivation in the book I am using is mathematically correct. The problem is about finding the volume integral of the gradient field. The author directly uses the Gauss-divergence theorem to relate the volume integral of gradient of a scalar to the surface integral of the flux through the surface surrounding this volume, i.e.
${\int <!--\; \int \; -->}_{CV}^{}\mathrm{\nabla <!--\; \nabla \; -->}\mathrm{\varphi <!--\; \varphi \; -->}dV={\int <!--\; \int \; -->}_{\mathrm{\delta <!--\; \delta \; -->}CV}^{}\mathrm{\varphi <!--\; \varphi \; -->}d\mathbf{S}$

Finding the volume of the pyramid and also determining the increase and decrease of its volume
The volume of a pyramid with a square base x units on a side and a height of h is $V={\displaystyle \frac{1}{3}}{x}^{2}h$.
1) Suppose $x=e^t$ and $h=e^{-<!--\; -\; -->2t}$. Use the chain rule to find $V^1(t)$.
2) Does the volume of the pyramid increase or decrease as t increases.

Given the following probability mass function, determine C.
Given the following probability mass function:
$P_{xy}(x,y)=C{\left(\frac{1}{2}\right)}^x\cdot <!--\; \cdot \; -->{\left(\frac{1}{2}\right)}^y$ determine C.
Hint: use the definition for a geometric series :
$\underset{n=0}{\overset{\mathrm{\infty <!--\; \infty \; -->}}{\sum <!--\; \sum \; -->}}{r}^n=\frac{1}{(1-<!--\; -\; -->r)}$
I'm not quite sure what to even do, I would normally integrate to find a constant? but I'm using geometric series now, and I don't really have any values for x,y or $P_{xy}(x,y)$...<br<x and y can take any integer value equal or bigger then 0

Find the maximum profit corresponding to a demand function of $p=36-<!--\; -\; -->4x$ and a total $\text{cost}\text{function}=2x^2+6$

Hello. Can you kindly help me solve this problem? Thank you in advance

Probability of weather on consecutive days.
Probability of a cloudy day is .55 Probability of a sunny day is .45
A)What is the probability of three consecutive cloudy days, followed by a sunny day?
B)What is the probability that exactly 1 out of 4 consecutive days will be sunny?
C)What is the probability that at least 1 out of 4 consecutive days will be sunny?
I think part A is a Geometric distribution where a success is a sunny day and the number of trials is 3.

The area of a trapezium is $475c{m}^2$ and the height is 19 cm .Find the lengths of its two paallel sides if one side is 4 cm greater than the other.

I'm not great at mathematics so I'm sure this is trivial to most. I have been searching around however and not been able to find how to figure out the incircle of a circle sector, or, in other words, the point inside a sector that is furthest away from the radii and the arc. The simpler solution the better

Find the function $s(x)$ such that $s(x)$ maximizes
${\int <!--\; \int \; -->}_0^{s^{-<!--\; -\; -->1}(k)}s(x)dx$
where $x\in <!--\; \in \; -->[0,10]$, $s(x)\in <!--\; \in \; -->[0,1]$, and $k\in <!--\; \in \; -->[0,1]$ ($k$ is a constant).

If argument of $\frac{z-<!--\; -\; -->z_1}{z-<!--\; -\; -->z_2}$ is $\frac{\mathrm{\pi <!--\; \pi \; -->}}{4}$, find the locus of $z$.
$z_1=2+3i$
$z_2=6+9i$
Approach: I tried to solve the equation using diagram, basically plotting the points on the Argand plane. What I got is a circle with center $7+4i$ and a radius of $\sqrt{26}$ units. The two complex numbers given lie on this circle, and form a chord. Any point lying on the major arc of this chord satisfies the condition.
How exactly would I represent this as a locus of the point? And is there any other method that I can use that does not involve a diagram?