0
votes
1answer
9 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
16 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 ...
1
vote
0answers
18 views

3-Point Shoot using Quadratic Equation [on hold]

This is my assignment. The question is "In what part of the three-point line can a player do best the three-point shoot to gain 3 point but using quadratic equation." There are no data given but we ...
4
votes
1answer
37 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
1answer
142 views
+50

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
45 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
38 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
95 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
27 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
25 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
43 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
97 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
55 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
75 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
53 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
47 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\\ ...
0
votes
2answers
24 views

Having a bit of trouble with min/max distance from sphere to point

The sphere is $x^2 + y^2 + z^2 = 81$ and the point is $(5,6,9)$ I am using Langrane multipliers , but the answers I am getting are so far off. I will post my system of equations soon. I found ...
-1
votes
1answer
62 views

How to get the third point coordinates in isosceles triangle?

Isosceles triangle $ABC$ $AB = AC = d_1$ $BC = d_2$ $A = (x_1, y_1)$ $B = (x_2, y_2)$ $C = (x_3, y_3)$ $\angle BAC = \phi$ $\angle ABC =\angle ACB = \theta$ I want an equation for $x_3$ and $y_3$ ...
1
vote
2answers
66 views

Can anyone prove that this is an envelope of a parabola?

Based on my last question I learned that this is an envelope of a parabola What is this geometric pattern called? But how can I prove it ?
0
votes
2answers
16 views

Problem about moving sides of triangle

Imagine a triangle XOY which sides lie on x-axis and y-axis with hypotenuse XY of length 5 m. Suppose the point X moves away from the (0,0) along x-axis with speed = 1 m per second. What speed the ...
-1
votes
1answer
60 views

How to find the volume of a solid [closed]

Suppose the solid $E$ is given by $$E=\{(x,y,z): (x^2+y^2+z^2+8)^2 \leq 36(x^2+y^2)\}.$$ Find the exact volume of $E$.
21
votes
5answers
4k views

Why is $\sin(d\Phi) = d\Phi$ where $d\Phi$ is very small?

I haven't touched Physics and Math (especially continuous Math) for a long time, so please bear with me. In essence, I'm going over a few Physics lectures, one which tries to calculate the Force ...
2
votes
0answers
75 views

What if the cow could fly?

See grazing cow. Now keep the restriction that the length of the rope is $l\leq\pi r$ where $r$ is the radius of the barn, (I like to think of this as a goat tied to a silo) but now suppose the cow ...
0
votes
3answers
101 views

Area of the field that the cow can graze.

How do we find the area that the cow can graze? The question goes as follows-- There is a circular barn house surrounded by a huge grazing field. A cow is tied to the rope ($AB$) at the end $A$ as ...
1
vote
1answer
27 views

Why does this get the angle of the surface?

I have this (physics) question, but am missing something as to why the math works for it. The problem is as follows: A 4- kg sphere rests on t he smooth parabolic surface. Determine the normal ...
0
votes
1answer
19 views

Determine Center Point based on 2 separate elipses

First timer here. I've been digging back into my good old maths days but am extremely rusty (beyond belief). I got a really tricky question that i want to determine formula for so that my mate can ...
4
votes
3answers
282 views

What does the secant value represent?

What does the secant value represent? I know that $$\sec = 1/\cos(\theta)$$ but really I do not know what this value represents, so I need your help. A clear example with images would be appreciated. ...
-1
votes
1answer
53 views

what does secant mean in mathematics?

I need your help. pleas I need some one to explain for me what does secant mean in mathematics? Clear example with images would be appreciated.
1
vote
1answer
53 views

How to calculate the number of lattice points in the interior and on the boundary of these figures with vertices as lattice points?

We define a point $(x,y)$ in the plane to be a lattice point if both $x$ and $y$ are integers. Now let $$S\colon= \{ (x,y) \ | \ 0 \leq x \leq m, \ 0 \leq y \leq \frac{nx}{m} \}, $$ where $m$ and ...
1
vote
0answers
31 views

How to establish the formula for area of a triangle using the axioms of area?

We have the following definition: AXIOMATIC DEFINITION OF AREA We assume there exists a class of $M$ of measurable sets in the plane (i.e. subsets of the plane whose area can be defined) and a set ...
0
votes
1answer
32 views

The surface area of a ring: $\pi[(r+dr)^2 - r^2]$ or $2\pi r\,dr$?

I know this may be really simple but here it is nonetheless. Let's say that I have a ring with a radius of $r$ and width of $dr$. I'm trying to find the surface $dS$ of the ring. Isn't it $dS = ...
0
votes
2answers
28 views

These two definitions of a hyperplane are equivalent?

My book first defines a hyper plane in $R^n $ as set $H= \{p^tx=\alpha \} $ where $p $ is a nonzero vector and $\alpha $ is scalar, or equivalently as the set $H=\{x:p^t(x-\bar{x })=0$. Next it ...
0
votes
0answers
17 views

How to find iso function value points without exploring all points in 2D space

Consider a 2D graph with dim1 and dim2 represented as X and Y respectively. The range of X and Y are 1 to 100. Hence there are 10000 points in the 2D space. Each point in the space is some function of ...
0
votes
1answer
27 views

how to find iso-cost contours on a 2d plot efficiently

Consider a 2D plot in which dimension 1 and 2 represent quantity 1 and 2 respectively ranging over 0 to 100. Each point in the space corresponding to (x,y) represent cost of choosing quantity 1 as x ...
1
vote
2answers
20 views

Equation of the plane that passes through a line and it's parallel with another line

What is the equation for plane $P$ that passes through $D_1:( x = 1+2t, ~y=t,~ z=t )$ and its parallel with $D_2 : ( x = -t,~ y=t, ~z = 2+3t )$? I'm just learning this. I think that if $P$ plane ...
1
vote
1answer
24 views

Transformation shifts parallelogram to trapezoid - fairly simple

We are given the region $D= {\{(x,y) | 1 \leq x-y \leq 2, x \leq 0, y\leq 0\} \subseteq \mathbb R^2}$ I drew this region on a piece of paper, it resembles an infinite parallelogram on the third ...
0
votes
0answers
29 views

Geodesics of a cone

To find Geodesics on a cone I used the cylindial coordinates $x=rcos\theta$ $y=rsin\theta$ $z=z$ Is this parameterization correct.How can I know how to parameterize? Then arc length $ds^2=r^2 ...
9
votes
2answers
430 views

Why is a straight line the shortest distance between two points?

The first application I was shown of the calculus of variations was proving that the shortest distance between two points is a straight line. Define a functional measuring the length of a curve ...
1
vote
1answer
53 views

Calculate minimum width of lane

A lane runs perpendicular to a road $64 ft$ wide. If it is just possible to carry a pole $125 ft$ long from the road into the lane, keeping it horizontal, then what should be the minimum width of the ...
0
votes
1answer
48 views

$ \frac{dy}{dx} = \tan(a) $ ; the derivative of a circle at a point (tangent) with respect to $y$ and $x$

warm up derivative: slope of the tangent line for the top half of a circle. I found that $$ \dfrac{dy}{dx} \left( x^2 + y^2 = 1 \right) \\ \rightarrow \dfrac{dy}{dx} = -\dfrac{x}{y} $$ please excuse ...
1
vote
1answer
33 views

surface area of the graph of a convex function

I started out with the following question: Say $\Omega$ is a nice bounded domain in $\mathbb{R}^{n-1}$. (One can imagine it being a unit ball in $\mathbb{R}^{n-1}$.) Let $f:\Omega\rightarrow ...
0
votes
2answers
22 views

Finding the equations of surfaces of revolution

I have the following question: $$\text{Sketch and find the equations of the surfaces formed by}$$ $$\text{i) }x^2 - y^2 + 1 = 0 \text{ about the y-axis}$$ $$\text{ii) }x^2 - 2y^2 + 2a^2 = 0 \text{ ...
7
votes
3answers
141 views

Beginning of Romance

I am a 17 guy from India. The fascination of maths has struck me recently, while I am in standard 12th. But all the resources I have, is some school textbooks. M.L Agrawal's of 11th and 12th. I don't ...
4
votes
1answer
64 views

Non-brute-force proof of parabola tangent property

I'm working through a classic Calculus book (Morris Kline), and one of the problems is: Prove that the foot of the perpendicular from the focus to any tangent of a parabola lies on the tangent ...
2
votes
1answer
44 views

How to reason about two points on the unit sphere.

I've recently been thinking about various problems involving two points on the surface of a unit sphere. Let's specify them with a pair of unit 3-vectors ${\bf \hat a}$ an ${\bf \hat b}$. Is there ...
1
vote
2answers
46 views

The rate change of the radius of a coil.

Suppose I have a tube of radius $r_0$ that I want to wrap a sheet of length $l$ and thickness $\Delta x$. Assuming the radius changes only when the paper overlaps the where the previous section ...
0
votes
3answers
69 views

geometric interpretation of Taylor's theorem

Deriving Taylor's theorem is not a problem. But I am curious if there is any nice geometric interpretation of the theorem.
0
votes
1answer
29 views

Calculte new width and height of the video based on the original width, height and ratio

after searching through - I wasn't able to find the answer so I'll give it a shot here. I need to resize desktop video in order to feet it on mobile screen, let's say original width of the video was ...
1
vote
1answer
78 views

The shortest path connecting three points

I have 3 points X,Y,Z, lets call them buildings. I need to find the shortest amount of path that connects the 3 buildings, these buildings can be in any sort of shape and any distance from each ...