0
votes
1answer
33 views

Geometric meaning of $z=\frac{x_1+x_2}{2}$, $y=\frac{x_1+x_2}{|x_1+x_2|}$, and of a substitution (in the complex plane)

Let $S$ be a unitary (that is, $r=1$) circle centered in the origin of the complex plane. Let $x_1 \in S$ and $x_2 \in S$ be the vertices of the triangle $Ox_1x_2$. What is the geometric meaning of ...
0
votes
0answers
32 views

Limits, tangents and areas. Why are these statements intuitive?

I'm reading Calculus: Early Transcendentals, by Anton, Bivens and Davis (9th edition). I'm not understanding a few things. Could someone please help me? On page 68 it says: "suppose that we are ...
0
votes
0answers
21 views

Determine sine wave frequency from two arbitrary points

If I have only two arbitrary points on a sine wave, what would be the simplest method for determining the frequency of the sine wave? The frequency is unknown. The bandwidth is restricted, the time ...
3
votes
3answers
322 views

Surface area of a sphere limits

If I am finding the surface area of a sphere in spherical coordinates my intergral would be like this: $$\int^{\pi}_0 \int^{2\pi}_0 R^2 \sin (\theta) d\phi d\theta =4\pi R^2$$ But if I do the ...
0
votes
1answer
25 views

Discuss the following graphs(Differential Equations)

So I have a differential equations midterm coming up soon, and in my last exam I messed the graphing question up. It was very similar to the one I am posting. All the questions said was "Discuss the ...
2
votes
2answers
50 views

Minimum volume cone.

What would be the radius and the altitude of a right circular cone that circumscribes a sphere with a radius 8 cm if the volume of the cone is to be minimized? Here is my rough sketch; My idea is ...
0
votes
0answers
16 views

interpreting the wording and geometry in an integration problem

The base of a solid S is a circle of radius 4. Cross-sections perpendicular to a given diameter are isosceles right triangles with the hypotenuse lying in the base. Based on the wording of the ...
2
votes
1answer
45 views

Curvature kappa with known acceleration, unit normal and unit tangent vectors

The acceleration of a particle is $a(t) = (4\sin t \cos t)T(t) + (4e^t \sin^2 (t/6))N(t)$, where $T(t)$ is the unit tangent vector and $N(t)$ is the unit normal vector. At $t = \pi/2$, the speed of ...
0
votes
1answer
49 views

Symmetry in mathematics

Why does maxima occur mostly at equality with a fixed condition for geometrical problems like in sum of sines, sum of cosines (considering a triangle), and also in problems like finding maximum area ...
0
votes
1answer
29 views

Possible length of isosceles triangle side.

The perimeter of a right triangle $RST$ is equal to the perimeter of isoceles triangle $xyz$ The lengths of the legs of the right triangle are 6 and 8. If the length of each side of the isoceles ...
6
votes
7answers
396 views

Is there a geometrical definition of a tangent line?

Calculus books often give the "secant through two points coming closer together" description to give some intuition for tangent lines. They then say that the tangent line is what the curve "looks ...
0
votes
0answers
27 views

is there an existing formula in finding the area of a rhombus wherein only the side is given?

is there an existing formula in finding the area of a rhombus wherein only the side is given? No measure of angles, no lengths of diagonals , height, etc. is given.
0
votes
2answers
8 views

node in quadrilateral

Let $(x_1,y_1),(x_2,y_2),(x_3,y_3),(x_4,y_4)$ be the vertices of a quadrilateral. How could I know that an arbitrary point $(x_5,y_5)$ is inside this quadrilateral or outside? Is there any formula to ...
2
votes
0answers
56 views

Least sum of power of distances

Let $n$ points in a $3$-dimensional space. Find the point $X$ that minimizes the sum of distances $\|A_1X\|^q+ \|A_2X\|^q + ... +\|A_nX\|^q $ (where $q \in \mathbb{Q}$). Are there any general ...
0
votes
0answers
22 views

Help me understand chained rotations

Ok, so for my thesis I am trying to do some stuff with ellipsoids in 3-dimensional space. I am trying to rotate an ellipsoid to face a certain direction using Tait-Bryan chained rotations. That is, ...
1
vote
0answers
25 views

Finding a function with desirable behaviour

I was playing around with math in my spare time and came up with the following problem. Suppose I have a $1$-parameter family of lines in the plane. Each line is given by the following equation ...
4
votes
2answers
169 views

Least sum of distances

Problem: Let $A, B, C, D$ be points in a $3$-dimensional space. Find the point $X$ that minimizes the sum of the distances $AX+ BX + CX + DX$. Context: During a course, I was assigned a ...
11
votes
6answers
351 views

Book with novel approaches to analysis

Now I'm studying Rudin's Principles of mathematical analysis, but I'm searching for a book that offers geometric, physical or otherwise non-standard approaches to topics in analysis. Also, I'm looking ...
0
votes
1answer
91 views

Infinitely Many Circles in an Equilateral Triangle

In the figure there are infinitely many circles approaching the vertices of an equilateral triangle, each circle touching other circles and sides of the triangle. If the triangle has sides of length ...
1
vote
1answer
35 views

slope of a curve in $\mathbb{R}^3$

The surface given by $z = x^2 -y^2$ is cut by the plane given by $y = 3x$, producing a curve in the plane. Find the slope of this curve at the point $(1, 3, -8)$. My answer is: $$f(x, y, z) = x^2 - ...
0
votes
1answer
50 views

Angle between tangent of Hyperbola and Xaxis

The problem is to find the angle between x axis and tangent to the hyperbola xy=9 at (3,3). The angle made by slope of the tangent of hyperbola at (3,3) is the angle between x-axis and tangent ?
0
votes
2answers
36 views

About the integral of the area in a circle

so i start with a circle with radius $r$, and make another one with radius $r+dr$, and the area between them is $(2\times \pi \times r)\times dr$ because that area is a rectangle, and that's where my ...
1
vote
1answer
65 views

Why geometric median cannot be solved analytically

$\DeclareMathOperator*{\argmin}{argmin}$ For a given set of $m$ points $x_1,...,x_m$ with each $x_i\in \mathbb{R}^n$, the geometric median (or the weber point) is defined as $$\argmin\limits_{y \in ...
1
vote
2answers
45 views

Distance of a Point from Hyperbola

Consider the part of hyperbola $H_{+}=\{(x,1/x)\colon x>0\}$ in the first quadrant, and $(a,b)$ any point in the plane (for sake of convenience, say $a,b>0$). If $(a,b)$ does not lie on the ...
0
votes
1answer
35 views

Surface integral over triangle

Given a triangle: $\{(x_1,y_1,z_1),(x_2,y_2,z_2),(x_3,y_3,z_3)\}$. How do I calculate the surface integral: $ I=\int\int \vec{F}\vec{n}dS$ with $ \vec{F} = [0,0,z] $, and where $ \vec{n} $ is the ...
1
vote
3answers
62 views

Calculate the depth of water in the trough when it is exactly half full

I am in my last year of high school and am currently studying for my finals by going over exercises in my coursebook. I came across this exercise and have been stuck on it for some weeks now. I have ...
0
votes
1answer
68 views

Differential and Integral calculus.

Can anyone here explain me, why do we take the Centre of mass of a conical shell using slant height and $dl$ whereas the centre of mass of a solid cone is calculated using the vertical height and ...
1
vote
2answers
160 views

A Norman window has the shape of a rectangle surmounted by a semicircle Problem

A Norman window has the shape of a rectangle surmounted by a semicircle. If the perimeter of the window is $24$ ft, express the area, $A$, of the window as a function of the width, $x$, of the window. ...
1
vote
1answer
79 views

Does a word problem provide all information?

A while ago I asked a similar question about word problems and assumptions. Is it a definition or an accepted-fact that word problems provide all information about the relevant existence/situation in ...
0
votes
1answer
30 views

Radius of curvature polar

$$ρ = a sin^3({\theta/3}) , ρ=\sqrt{(x^2+y^2)}$$ Please help me find the radius of curvature of this problem I have solved parametric and polynomial forms but i am unable to get this
0
votes
0answers
27 views

Radius of curvature

I want to find the radius of curvature of this equation ρ = a sin θ I have solved problems of courvature about polynomial equations and parametric equations but i am not able to understand the ...
3
votes
2answers
51 views

The length of the smallest possible ladder to change the bulb.

In the drawing, the point P, a bit located under the bulb, has coordinates (a, b​​), where a and b are two parameters. You want to change the bulb, and for this it is necessary to install a ladder ...
2
votes
1answer
35 views

Show that this construction is a parallelogram.

Let $ABC$ be a triangle. The middle of the segment $BC$ is denoted by $M$ and the centroid of $ABC$ is rated $G$. We construct $G'$ on the line $GM$ such that $|GM|=\frac{1}{2}|GG'|$ and ...
1
vote
0answers
26 views

Distance of the plane relative to the base of a Pyramid.

Consider a pyramid is cut by a plane parallel to its base. Question: What is the distance of the plane relative to the base so that the volume of the truncated pyramid so formed is $\frac{3}{8}$ of ...
2
votes
1answer
12 views

Angles of which locations of points corresponding to distances intersect each other

Original problem : In the $XY$ plane,Let $P_1$ and $P_2$ two points with coordinates $(-1,0)$ and $(1,0)$. $C_1$ is the locus of points whose sum of distances to $P_1$ and $P_2$ is $4$. $C_2$ is the ...
3
votes
1answer
68 views

inflection point of cubic bezier with restrictions

Say you have this type of cubic Bézier curve: The 4 control points A,B,C,D have restrictions: A & B have the same Y-axis coordinate C & D have the same Y-axis coordinate B & C have ...
4
votes
1answer
39 views

Locating a radar in a plane

Given two located targets at $(x, y)=(- 2.0)$ and $(x, y)=(2,0)$. A radar, located in an unknown location of the $XY$ plane, and sends a pulse and in return receives pulses from the two targets. ...
7
votes
2answers
213 views

Hands of the clock, Revisited.

It has already been answered (here) that it is impossible for the (continuously moving) hands of a clock to trisect the face of said clock. Even ideally the hour, minute, and second hand can never ...
2
votes
1answer
59 views

Application of Bessel Function

I have read number of books and online literature on Bessel function. Theoretically, I have known about Bessel function. What is practical significance of Bessel function? How can Bessel function ...
0
votes
2answers
40 views

Area such that $d(P, AB) \leq \min \{d(P,BC), \ d(P,AC)\}$

$d(P,L)$ means distance of point $P$ from line $L$. There are three points $A(4,4)$, $B(8,4)$ and $C(4,6)$. We need to find the area of the region satisfying $d(P,AB) \leq \min \{d(P,BC), ...
2
votes
2answers
111 views

The area of the region $|x-ay| \le c$ for $0 \le x \le 1$ and $0 \le y \le 1$

What is the area of the following region: $|x-ay| \le c$ for $0 \le x \le 1$ and $0 \le y \le 1$. Assume $c>0$. We can also assume $|a|>1$ since it will not change the essence of the ...
0
votes
0answers
31 views

Volume of a rotated regular polygon

I want to calculate the volume of the shape which is created when you rotate a regular $n$-sided polygon around the $y$ axis with a major radius $r$. (like a torus, but with a polygon as rotated ...
1
vote
0answers
27 views

Slicing through a cuboid containing spheres, how many are exposed to the surface and what is their combined volume

So I place spheres of radius chosen at random from a normal distribution of known mean and standard deviation in a cub or cuboid at random (not overlapping) until a known density of the entire cube is ...
3
votes
1answer
49 views

Maximum area of quadrilateral with sides and diagonals not exceeding a given value

What is the maximum area of quadrilateral such that max|sides, diagonals|< x ? I know that the maximum area would be the square area of diagonals of length $x$ which has area $x^2/2$, but how ...
4
votes
3answers
101 views

What's the geometrical meaning of the derivative of $e^x$

That is to say, is there a geometrical reason why the derivative of $e^x$ is $e^x$ itself?
3
votes
1answer
56 views

Is there a geometric interpretation for lower order terms in sum of squares formula?

So we know that $\sum_{k=1}^n k^2 = n(n+1)(2n+1)/6$ and if we think about making a square layer out of $n^2$ unit cubes, and then placing a square layer of $(n-1)^2$ unit cubes on top of the first ...
0
votes
2answers
45 views

Area of intersection of two regions formed by

what is the area of intersection of $a<\frac{x}{y}<b$ and $c<\frac{y}{x}<d$? Thank you very much
1
vote
3answers
87 views

Area of the polygon formed by cutting a cube with a plane

I want to determine the area of a polygon that is formed when a cube is cut by a plane like shown below: Where the blue triangle is in fact a part of the plane described by $$m_x x + m_y y +m_z z ...
2
votes
1answer
58 views

Arc length inequality on ellipse

Suppose $E$ is the ellipse $(\frac{x}{a})^2+(\frac{y}{b})^2=1.$ For $P,Q\in E$ let $\sigma(P,Q)$ denote the length of the minor arc connecting $P$ to $Q$. Find the maximum value for $A>0$ such that ...
0
votes
2answers
50 views

Solve nonliner equations

We are trying to find intersection of hyperbolas and we ended up in five equations $$\begin{align} A_1X^2+B_1Y^2+C_1XY+D_1X+E_1Y+F1&=0\\ A_2X^2+B_2Y^2+C_2XY+D_2X+E_2Y+F2&=0\\ ...