All Questions
214
questions
3
votes
3
answers
306
views
Finding the radii that maximizes and minimizes the area of four inscribed circles in an equilateral triangle.
An equilateral triangle with side length $1$ unit contains three identical circles $C_1$, $C_2$ and $C_3$ of radius $r_1$, each touching two sides of the triangle. A fourth circle $C4$ of radius $r_2$ ...
3
votes
1
answer
112
views
Ratio of Perimeter^3 to the Area of an Isoceles Triangle.
I am in trouble with the following question:
QUESTION
ABC is an isosceles triangle inscribed in a circle of radius $r$. If $AB=AC$ and $h$ is the altitude from $A$ to $BC$ and $p$ be the perimeter ...
- 772
3
votes
1
answer
270
views
What does Spivak want me to do?
This goes on in Chapter 8, on least upper bounds and related topics. I have proven $(a),(b),(c)$.
The sketch is.
$(a)$ If $\{a_n\}$ is a sequence of positive terms such that $$a_{n+1}\leq a_n/2$$ ...
- 123k
3
votes
1
answer
197
views
Plotting a fix number of points evenly across surface area of a circle or sphere.
I would like to design an LED sphere, but I am having some trouble deciding on the placement of LEDs evenly across its surface area.
I would like there to be 32 evenly spaced LEDs across the ...
- 33
3
votes
0
answers
181
views
Question about the second derivative of a circle
I am trying to find the second derivative of a general circle but I can't seem to get the right answer.
My working goes as follows:
$$
(x-a)^2+(y-b)^2=R^2
$$
$$
2(x-a)+2(y-b)*\frac{dy}{dx}=0
$$
$$
\...
- 417
3
votes
1
answer
297
views
Newbie: Find the intersection of a line and a circle and interpret geometrically
Find the points of intersection of the line $x+y+k=0$ and the circle $x^2+y^2=2x$. Show that there are two points of intersection if: $-1-\sqrt2<k<-1+\sqrt2$, one point of intersection if: $k=-1\...
- 43
2
votes
5
answers
3k
views
How to derive formula for circumference of circle using integration?? [closed]
Can using the formula of arc of a curve be of use here...as in:
$$L = \int_{{\,a}}^{{\,b}}{{\sqrt {1 + {{\left( {\frac{{dy}}{{dx}}} \right)}^2}} \,dx}}$$
Maybe we could consider the semicircle, find ...
- 47
2
votes
3
answers
5k
views
Use calculus to derive area of circle using n triangles
This is a homework question I am struggling with...
Let $n$ be a positive integer, and cut the circle into $n$ equal sectors. In each sector there is an isosceles triangle formed where the edges of ...
2
votes
2
answers
149
views
Geometric Approximation for Area of Circle Using Calculus
A couple of years ago, I came up with this formula:
$$\lim_{n\to\infty}\frac{180\left(\pi^2r^2\cot\left(\frac{180}{n}\right)\right)}{n\pi}=\pi r^2$$
I derived it from a geometric perspective and just ...
- 1,030
2
votes
2
answers
239
views
working backwards from $\pi r^2$
I have been dipping my toes into a bit of calculus (through the better explained website), however I have become stuck on my understanding of the area of a circle. I understand that the formula for ...
- 177
2
votes
2
answers
1k
views
Integral of the Product of a Function and Its Derivative
Problem: Let $f$ be a function such that the graph of $f$ is a semicircle $S$ with end points $(a,0)$ and $(b,0)$, where $a < b$. The improper integral $\int_{a}^{b} f(x) f'(x) dx$ is
(A) ...
- 8,040
2
votes
2
answers
322
views
Circumference of a ring
Circumference of a Circle is $2 \pi r$ , what about a ring though ? It will have two radiuses : Which one will we take into consideration while measuring the perimeter ? Will it have two different ...
- 135
2
votes
2
answers
550
views
Geometric proof of chain rule with the derivative of $\sin(2x)$
I'm following this post https://math.stackexchange.com/a/2169/612996 as my example and I've figured out how it works for $\sin(\theta)$,
During my first try: I keep on missing the factor of $2$ when ...
user612996
2
votes
2
answers
1k
views
circular reasoning in proving $\frac{\sin x}x\to1,x\to0$
The classic proof for $\frac{\sin x}x\to1,x\to0$ is using a squeezing theorem based on arguments about areas of circles.
But as far as I know, all deduction of formula of circles' area is based on ...
- 21
2
votes
2
answers
78
views
Surface area - even dimensions
Consider the surface:
$$ (\log x_1)^2+(\log x_2)^2+\cdot\cdot\cdot +(\log x_n)^2=R $$
For $R=1$ and in even dimensions $n=2,4,6, \cdot\cdot\cdot$ we have the volumes:
$$V=\bigg( \frac{\pi^2 I_1(\sqrt{...
- 756