0
votes
1answer
27 views

How to get the third point coordinate 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
61 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
59 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
72 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
90 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
25 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
280 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. ...
0
votes
1answer
51 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
40 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
29 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
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
15 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
26 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
19 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
23 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
26 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
235 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
47 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
29 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
21 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{ ...
6
votes
3answers
129 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
62 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
43 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
63 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
25 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
75 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 ...
1
vote
1answer
49 views

Griffiths Electrodynamics Example 1.8 - Calculating Volume Integral

In Example 1.8 in the electrodynamics textbook by Griffiths, he calculates the volume integral over a prism. The prism is formed of two triangles in the xy plane, with sides $x=0$ to $x=1$, $y=0$ to ...
0
votes
2answers
40 views

Calculate height or width of rectangle given its area and height or width.

I guess this should be very easy, but I'm stuck. I have a rectangle: Area = $520\text{ cm}^2$ $x = 120\text{ cm}$ $y = ?$ So: Area = $x \cdot y$ $520 \text{ cm}^2 = 120\text{ cm} \cdot y$ And ...
0
votes
1answer
36 views

Cylindrical hoof lateral surface and volume

I am trying to figure out the proof for lateral surface and volume of a cylindrical hoof (or, more generally, a cylindrical wedge) given by Wolfram MathWorld but I am having some trouble understanding ...
3
votes
1answer
51 views

Equivalence of the two cosine definitions

There are at least two ways to define the cosine function: You can define it with a right triangle in the unit circle and extend the definition to $\mathbb{R}$. (classic definition) The other ...
4
votes
2answers
70 views

2 calculus questions with integration - check me

I have 2 questions I would like assistance with. 1) Find the area of the region bounded by the graphs $y=5x, y=15x, y=\frac{4}{x}, y=\frac{8}{x}$ This was very difficult and tedious. I had trouble ...
0
votes
1answer
60 views

Find the equation of a plane which is perpendicular to another plane

Find the equation of a plane which is perpendicular to the plane $\pi\equiv x+2y-2z+3=0$ and it intersects it through the line that lies in the XOZ plane. Normal vector of the given plane is ...
1
vote
2answers
67 views

Find the axes of the ellipse

What are the equations of the major and minor axes of the ellipse $x^2+2y^2-2xy-1=0$. The centre of the ellipse is $(0,0)$ but the axes are tilted (with respect to $x-y$ axes). I don't know how to ...
2
votes
2answers
34 views

Finding the equation of tangent line

I'm stuck with problem supposed to be trivial. I need to find tangent line witch touches curve $y^2 = -4ax$ at the point $(x_0,y_0)$ Rewriting it as $$x = -\frac {y^2}{4a}$$ Taking derivative: ...
0
votes
0answers
34 views

The Least Characteristic of Shapes in $\mathbb{R}^n$

Fix the following notations: "Shape" denotes a closed curve in $\mathbb{R}^2$ or a closed surface in $\mathbb{R}^3$. $P$ denotes the circumference of a shape in $\mathbb{R}^2$. $A$ denotes the area ...
0
votes
1answer
44 views

Volume of a square pyramid— what's wrong in my analysis

Note: Its not a duplicate. The other one talks about triple integrals... while I am only doing a single one..! So to get the area of a cube I can do the following integral, $$\int_0^a a^2 \, dh = ...
9
votes
4answers
276 views

The 'sine and cosine theorem' - formulas for the sum and difference

I've read somewhere that the sine and cosine functions can be fully described by this theorem: $\sin(0) = 0, \cos(0) = 1$ $\sin(a-b) = \sin(a)\cos(b) - \sin(b)\cos(a)$ $\cos(a-b) = \cos(a)\cos(b) + ...
1
vote
0answers
45 views

Wedge volume of sphere problem

The Clare College bridge at Cambridge is decorated with 14 stone spheres, but one of it missed a wedge. I took a photo to estimate the volume of the missing part of the sphere. I am not confident ...
0
votes
1answer
14 views

How to Do Trilateration?

Trilateration is the process of calculating the coordinates of a point by using its distances to three other points. Say that, we have three points of which we know the coordinates: $A(A_x, A_y)$ ...
0
votes
3answers
57 views

Find the locus of points

Find the locus of points, the distance between them and the point $(2,1)$ is equal to the distance between them and the straight line $4x = 3y$ I know that it is the definition of a parabola But I ...
0
votes
1answer
51 views

Curl and unit vectors after a change of coordinates

Let $x,y,z$ be a system of coordinates with unit vectors respectively $\mathbf{u}_x$, $\mathbf{u}_y$, $\mathbf{u}_z$. Moreover, we have $\nabla = \displaystyle \frac{\partial}{\partial x} ...
2
votes
3answers
390 views

How to calculate this shape's volume

So I've got this shape How would I calculate the volume? I thought about splitting it up into a cone somehow but I don't have the rest of the information to do that, I think...What's to do? This ...
1
vote
1answer
25 views

Prove uniqueness of polar-coordinates $(R>0, \theta)$ up to angle $\theta+ 2 \pi$.

Prove uniqueness of polar-coordinates $(R>0, \theta)$ up to angle $\theta+ 2 \pi$. Suppose we have $(x,y) \in \mathbb R^2$. Then we can transform this point to polar-cordinates $(R>0, ...
0
votes
2answers
111 views

Optimization of the surface area of a open rectangular box to find the cost of materials

A rectangular storage container with an open top is to have a volume of 10 cubic meters. The length of the box is twice its width. Material for the base costs ten dollars per square meter and for the ...
1
vote
0answers
17 views

Integrating along a line of point sources

I have some concentration that radiates from a spherical point, being steadily consumed until it hits zero at some distance $r_{n}$. This is given by $$C(r) = A\left[r^2 + \frac{2r_{n}^3}{r} - ...