Expert Help with High School Geometry

The probability of the dart hitting the smaller ring is?
A dart is randomly thrown at a circular board on which two concentric rings of radii R and 2R having the same width (width much less than R) are marked. The probability of the dart hitting the smaller ring is:
(1) Twice the probability that it hits the larger ring.
(2) Half of the probability that it hits the larger ring.
(3) Four times the probability that it hits the larger ring.
(4) One-fourth the probability that it hits the larger ring.

What is the distance between (4,2,6) and (7,3,6)?

Volume of a Solid of Revolution?
Find the volume if the region enclosing $y=x^3,x=0,$, and $y=8$ is rotated about the given line. So the axis of rotation that was given was the y axis, but I'm a bit confused on what the given bounds of $y=8$ and $x=0$ have to do with it, but I pretty much ignored them and put the given equation in terms of y and used the integral ${\int <!--\; \int \; -->}_0^7(2^2-<!--\; -\; -->(\sqrt[3]{y}{)}^2)$ to solve, is this at all right? Any help would be appreciated Also looking for some guidance on how I'd go about finding the volume if the axis of rotation was something like $x=6$.

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

Conceptual doubt in taking elements for integration
In physics, we were taught to find the center of mass using calculus for different shapes. I tried to find the volume of a sphere using similar concepts.
I took a cylindrical element to integrate whose radius is $R\sin <!--\; \; -->\mathrm{\theta <!--\; \theta \; -->}$ where $\mathrm{\theta <!--\; \theta \; -->}$ is the angle made with the vertical to the edge of the element . And took the height to be $Rd\mathrm{\theta <!--\; \theta \; -->}$ when I integrated all the cylindrical elements varying theta from o, $\mathrm{\pi <!--\; \pi \; -->}$ and did not get the right answer.
I realize the error is taking $Rd\mathrm{\theta <!--\; \theta \; -->}$ as height but why is it that we can take the "extension" to be $rd\mathrm{\theta <!--\; \theta \; -->}$ when finding the surface area of the sphere.
To state my question more clearly:
Why cant the height of the element be $Rd\mathrm{\theta <!--\; \theta \; -->}$ when finding volume since it works when finding surface area.
What I did to find volume:
${\int <!--\; \int \; -->}_0^{\mathrm{\pi <!--\; \pi \; -->}}\mathrm{\pi <!--\; \pi \; -->}(R\sin <!--\; \; -->\mathrm{\theta <!--\; \theta \; -->}{)}^2(Rd\mathrm{\theta <!--\; \theta \; -->})$
What I did to find surface area:
${\int <!--\; \int \; -->}_0^{\mathrm{\pi <!--\; \pi \; -->}}2\mathrm{\pi <!--\; \pi \; -->}R\sin <!--\; \; -->\mathrm{\theta <!--\; \theta \; -->}(Rd\mathrm{\theta <!--\; \theta \; -->})$

Finding the volume using double integrals
Find the volume of the wedge sliced from the cylinder $x^2+y^2=1$ by the planes $z=a(2-<!--\; -\; -->x)$ and $z=a(x-<!--\; -\; -->2)$.
I am confused because $x^2+y^2=1$ is a unit circle not a cylinder. and the other two are lines in the zx plane where $y=0$. I don't see how they are planes, are they planes or are they just lines in the zx plane (when $y=0$).

Distance between (5,-6,4) and (-2,2,6)?

What is the expected volume of the simplex formed by $n+1$ points independently uniformly distributed on ${\mathbb{S}}^{n-<!--\; -\; -->1}$?

Let $X_n$ be a geometric random variable with parameter $p=\mathrm{\lambda <!--\; \lambda \; -->}/n$. Compute $P(X_n/n>x)$
$x>0$ and show that as n approaches infinity this probability converges to $P(Y>x)$, where Y is an exponential random variable with parameter $\mathrm{\lambda <!--\; \lambda \; -->}$. This shows that $X_n/n$ is approximately an exponential random variable.

How to solve this type of sum using indirect proof. For any real number $x$ , if $x^3+2x+33\ne <!--\; \ne \; -->0$ , then $x+3\ne <!--\; \ne \; -->0$

Triple integrals-finding the volume of cylinder.
Find the volume of cylinder with base as the disk of unit radius in the xy plane centered at (1, 1, 0) and the top being the surface $z=((x-<!--\; -\; -->1{)}^2+(y-<!--\; -\; -->1{)}^2{)}^{3/2}.$.
I just knew that this problem uses triple integral concept but dont know how to start. I just need someone to suggest an idea to start. I will proceed then.

Prove that the locus of the incenter of the $\mathrm{\Delta <!--\; \Delta \; -->}PS{S}^{\prime }$ is an ellipse of eccentricity $\sqrt{\frac{2e}{1+e}}$
Let S and S′ be the foci of an ellipse whose eccentricity is e.P is a variable point on the ellipse.Prove that the locus of the incenter of the $\mathrm{\Delta <!--\; \Delta \; -->}PS{S}^{\prime }$ is an ellipse of eccentricity $\sqrt{\frac{2e}{1+e}}$.
Let P be $(a\cos <!--\; \; -->\mathrm{\theta <!--\; \theta \; -->},b\sin <!--\; \; -->\mathrm{\theta <!--\; \theta \; -->})$. Let the incenter of the triangle PSS′ be (h,k). The formula for the incenter of a triangle whose side lengths are a,b,c and whose vertices have coordinates $(x_1,y_1)$, $(x_2,y_2)$, and $(x_3,y_3)$ is
$(\frac{a{x}_1+b{x}_2+c{x}_3}{a+b+c},\frac{a{y}_1+b{y}_2+c{y}_3}{a+b+c}).$
Then, $h=\frac{2c\cdot <!--\; \cdot \; -->a\cos <!--\; \; -->\mathrm{\theta <!--\; \theta \; -->}+P{S}^{\prime }\cdot <!--\; \cdot \; -->c-<!--\; -\; -->PS\cdot <!--\; \cdot \; -->c}{2c+P{S}^{\prime }+PS}\text{ and }k=\frac{2c\cdot <!--\; \cdot \; -->a\sin <!--\; \; -->\mathrm{\theta <!--\; \theta \; -->}}{2c+P{S}^{\prime }+PS},$
but I could not find the relationship between h and k, whence I could not find the eccentricity of this ellipse.

Volume Integration of Bounded Region
I'm trying to integrate this volume in spherical and cylindrical coordinates, but having difficulty finding my bounds of integration;
I'm given the region in the first octant bounded by $z=$ $\sqrt{x^2+y^2}$, $z=$ $\sqrt{1-<!--\; -\; -->x^2-<!--\; -\; -->y^2}$, $y=x$ and $y=$ $\sqrt{3}x$ and I need to evaluate ${\iiint <!--\; \iiint \; -->}_{V}dV$.
When proceeding to integrate with spherical and cylindrical coordinates I am not getting the right bounds such that both methods equate to the same volume? I am definitely missing something. Any and all advice would be much appreciated!

Finding the volume of oil in a cylinder which is lying parallel to the ground.
A cylinder of height 2 m and radius 2 m is partially filled with oil. When the cylinder is lying parallel to the ground the height of oil level is 1.5 m from ground. What is the volume of oil? How to solve it?

When A and B flip coins, the one coming closest to a given line wins 1 penny from the other. If A starts with 3 and B with 7 pennies, what is the probability that A winds up with all of the money if both players are equally skilled? What if A were a better player who won 60% percent of the time?

Distance between (3,6,2) and (0,6,0)?

Question of Triangle of Geometry
In triangle ABC,we have $AB>AC$. If A' is the mid point of BC, AD is the altitude through A and if the internal and external bisectors of angle A meet at BC at X and X' respectively ,prove that $A^{\prime }D=(c^2-<!--\; -\; -->b^2)/2a$
I have tried to use obtuse angle theorem to find height then solve the question but stuck on double variable.

The median salary of the 1,073 teachers in a large public school district is $38,850. If the 65th percentile of the salaries is $46,600 , find the percent of teachers whose salary is between $38,850 and $46, 600 .

Find the length of one side of a triangle given two other sides and the length of the angle bisector
For an arbitrary triangle ABC.
If given the length of side $AB=15$ and $AC=12$ and the length of the angle bisector that connects angle A to side $BC=10$. What is the length of BC.

Prove the formula for the foci of an ellipse
I have summarized the question below:
If the vertices of an ellipse centered at the origin are (a,0),(-a,0),(0,b), and (0,-b), and $a>b$, prove that for foci at $(\pm <!--\; \pm \; -->c,0)$, $c^2=a^2-<!--\; -\; -->b^2$.
I am guessing that I have to use the distance formula, which is $d=\sqrt{(x_2-<!--\; -\; -->x_1{)}^2+(y_2-<!--\; -\; -->y_1{)}^2}$.