Tagged Questions
0
votes
2answers
29 views
Perimeter or Calculus Word problem
A rectangular plot of farmland will be bounded on one side by a river
and on the other three sides by a single strand of electric fence.
With 1400m of wire at your disposal, what is the largest ...
3
votes
1answer
45 views
Volume of a hypersphere
We know that the area of a circle (2-D) =$\pi r^{2}$ and the volume of a sphere (3-D)= $\dfrac{4}{3}\pi r^{3}$.
Question:What is the "volume"(or whatever that is called) of a n-dimensional sphere?
...
0
votes
1answer
62 views
How to find a point on the tangent line whos length is 1?
im trying to figure out a formula to find the point(x,y) on a tangent line whos length is between 0 and 1 while it rotates around the unit circle uniformly, so the point would either be right on the ...
0
votes
0answers
61 views
Geometrical Inequality
Let $ABCD$ be a quadrilateral on the unit circle, and the diagonals
$AC$ and $BD$ intersects at $E$. If the shortest height of the
triangle $ACD$ equals the radius of the incircle of the triangle ...
5
votes
5answers
138 views
Does $e$ have a geometric representation? [duplicate]
Just like $\pi$ is the ratio of a circle's circumference to its diameter? I know that the tangent line to the function $e^x$ has a slope of $e^x$ at that point, but is there some other geometric ...
2
votes
2answers
80 views
Cheapest can problem
A cylindrical can which must hold 1000 mL is set to be designed so the least amount of material is necessary to make the can.
What should the radius be?
What is the height of the can?
What is the ...
1
vote
1answer
63 views
Question on the perimeter of any quadrilateral
Is it true that the perimeter of any convex quadrilateral inside a unit circle
is no more than $4\sqrt{2}$?
1
vote
1answer
82 views
Does there exist such a pentagon that can be covered by a circle?
Does there exist a pentagon in which every two nonadjacent vertices
is connected by a diagonal and the minimal height of every triangle formed by the
sides and diagonals of the pentagon whose two ...
7
votes
3answers
70 views
Proof by induction using Fubini's Theorem
I am asked for the volume of the region $x_1+\cdots+x_n\leq 1$ where $x_1,...,x_n\geq 0$. I am proposing that the volume $V(n)$, is given by
$$
V(n) = ...
3
votes
3answers
113 views
How can I find the points of intersection between the curves $r=1+\sin\theta$ and $r=1-\sin\theta$?
Find the points of intersection for the curve $r=a(1+\sin\theta)$ and $r=a(1-\sin\theta)$
My book says the answer is $(0,0),(a,0),(a,\pi)$.
However I calculated $ (a,0),(a,\pi),(a,2\pi)$.
1
vote
2answers
50 views
Volume of a spherical balloon question
How would I do this problem I am not sure if I did it correctly.
The volume $V$ of a spherical balloon is increasing at a constant rate of 32 cubic feet per minute.
A. How fast is the radius r ...
0
votes
3answers
58 views
Proofs on equilateral triangles
Let $\Delta$ be the set of all triangles with two equal edges and be inscribed in a circle of radius $R$.
So, how do I show that:
Equilateral triangle in $\Delta$ is maximizing the area?
and
this ...
5
votes
4answers
110 views
How to compute the area of the shadow?
If we can not use the integral, then how to compute the area of the shadow?
It seems easy, but actually not?
Thanks!
4
votes
1answer
87 views
Is this a valid proof of the derivatives of the trigonometric functions?
For the sake of this proof, the trigonometric functions $\cos$ and $\sin$ are defined as the coordinates of a point on the unit circle, rather than any of the modern analytic definitions.
Let $\vec ...
2
votes
1answer
27 views
Given 2 Sheets of Acrylic, determine the maximum dimensions of a rectangular prism.
I have been trying to figure this out all morning and I am having no success. I know I have to use calculus, but I am not sure exactly how to set this up.
I am trying to determine the maximum length ...
0
votes
1answer
26 views
A geometric curve problem
Prove an algebraic curve $y = {x-1 \over x^2 +1}$ has three simple inflexions and they lie on a same line.
0
votes
2answers
67 views
Surface area of an elliptic paraboloid
The elliptic paraboloid has height h, and two semiaxes a, b. How to find its surface area? Does it possible to use a direct formula without integrals?
0
votes
1answer
27 views
halfspaces question
How do I find the supporting halfspace inequality to epigraph of
$$f(x) = \frac{x^2}{|x|+1}$$
at point $(1,0.5)$
2
votes
1answer
90 views
Polar Coordinate Surface Integral
Been staring at this question for hours, to no avail..
Let $S$ be the paraboloid parametrised in polar coordinates as
$$t(r,x)=(r\cos \phi,r \sin \phi,r^2), \qquad r\geq 0, \quad 0\leq \phi\leq ...
2
votes
2answers
40 views
Manipulating the equation!
The question asks to manipulate $f(x,y)=e^{-x^2-y^2}$ this equation to make its graph look like the three shapes in the images I attached.
I got the first one: $e^{(-x^2+y^2)}\cos(x^2+y^2)$.
But I ...
2
votes
0answers
32 views
Kmeans on “symmetric” data
A set is said to be fully-symmetric if for every $x$ in it, negating one of its components results in $y$ such that $y$ is in the set as well.
A set is said to be semi-symmetric if for every $x$ in ...
1
vote
3answers
54 views
Trigonometry tangent line question
How would I figure this out.
Find all x values between $0$ and $2\pi$ where the line tangent to the graph of
$y=\frac{\cos x}{2+ \sin(x)}$ is horizontal.
I did the deriavative
...
0
votes
1answer
95 views
Find the shortest distance between $9x^2+9y^2-30y+16=0$ and $y^2=x^3$
Find the shortest distance between $9x^2+9y^2-30y+16=0$ and $y^2=x^3$.
I know the shortest distance exists between the curves on the common normal line. Is there any other shorter way to attempt?
3
votes
2answers
160 views
Find the maximum area of the regular pentagon
Find the maximum area of the regular pentagon that inscribed a unit square.
4
votes
2answers
169 views
Has anyone published a formula for the volume of the intersection of two balls
Math people:
I have Googled this question and searched Math Stack Exchange, and not found an answer. Given $r_1, r_0, t >0$, $r_0 \leq r_1$ I have found a formula for the volume of the ...
3
votes
3answers
102 views
Calculus word problem dealing with rectangle
How would I figure this problem out.
A rectangle has a base B and a height H. Assuming that is area remains constant express the rate of change of the height with respect to the base.
this is what I ...
1
vote
2answers
256 views
Surface area of a sphere?
How would I solve the following problem?
Find the rate of change of surface area of a sphere with respect to its diameter D.
I know the formula for surface area of a sphere is
$A=4\pi r^2$
So I ...
1
vote
2answers
101 views
Create asymmetric 3D function by revolving a 2D function around an axis
Could someone please let me know how I can construct the equation for asymmetric torus similar to the figure below? The asymmetric torus seems to be a 2D function revolved around an axis while being ...
1
vote
3answers
88 views
0
votes
2answers
189 views
Difference between a Gradient and Tangent
I am unable to understand the fundamental difference between a Gradient vector and a Tangent vector.
I need to understand the geometrical difference between the both.
By Gradient I mean a vector ...
1
vote
3answers
255 views
Area of the intersection of a cone and a sphere
Calculate the area of the surface $x^2+y^2 = z^2 $ with $z \geq 0$, limited by $x^2 +y^2+z^2=2ax$.
I think the method to solve it is to calculate the parametric equation of the curve and then ...
2
votes
1answer
151 views
Vector Geometry Proof
Hello, this problem states to prove that the line segments drawn from one vertex of the parallelogram to the midpoints of the opposite sides trisects the other diagonal. Only vector addition, ...
2
votes
1answer
65 views
What is the probability that the resulting four line segments are the sides of a quadrilateral?
Question: Divide a given line segment into two other line segments. Then, cut each of these new line segments into two more line segments. What is the probability that the resulting line segments are ...
2
votes
1answer
88 views
How can I find the curves in the problem?
It's the first exercise in this book (It has a drawing).
Let $ \rho = f(\theta) $ be the equation of a plane curve in polar coordinates. Through the pole $O$ draw a line perpendicular to the ...
2
votes
1answer
76 views
Rotation around slant line
What is the volume when $f(x) = x^2$ is rotated around the line $y = x$?
For each individual $x$, I was considering the difference between $\begin{pmatrix} x\\x \end{pmatrix}$ and the projection of ...
0
votes
3answers
215 views
Find a isosceles triangle with biggest plane area with perimeter 1
I'm trying to use Pythagoras. Assuming $ a=b, v = 2a + c $ I tried calculating height (Vc) on c. Vc by expressing it with a & c. And then using one of the variables a or c in a function to ...
1
vote
1answer
139 views
Method of Exhaustion applied to Parabolic Segment in Apostol's Calculus
In section I1.3 of Apostol's Calculus (2nd Ed., Vol. 1, pages 3 to 7), Apostol details how to apply the method of exhaustion to a the parabolic segment of $x^{2}$. I understand the process of applying ...
5
votes
1answer
189 views
Proof of pythagorean theorem
Any one seen this proof before?
$$\frac{d}{dx} \sin(x)^2=2\cos(x)\sin(x)$$
$$\frac{d}{dx} \cos(x)^2=-2\cos(x)\sin(x)$$
$$\frac{d}{dx} \sin(x)^2+\frac{d}{dx} \cos(x)^2=0$$
$$\sin(x)^2+\cos(x)^2=c$$
...
8
votes
2answers
294 views
Optimization of the area of a cross inscribed in a circle
I've really been scratching my head over this optimization problem. "Consider a symmetric cross inscribed in a circle of radius $r$." The length from the center of the cross to the middle of one of ...
0
votes
2answers
83 views
Area of $2^n$-gon inscribed in a Unit Circle
I came across the following problem:
Find the area of a $2^n$-gon inscribed in a circle and rigorously prove that the area tends to $\pi$ as $n\to\infty$. The area is easily shown to be $$ ...
10
votes
2answers
236 views
What Mathematics questions can be better solved with concepts from Physics?
Over the years, I've seen several questions in mathematics that can be solved using concepts borrowed from Physics. Having seen these question, I'm interested to find out what other mathematics ...
1
vote
1answer
169 views
Maximum area for fixed perimeter of a triangle
I'm trying to prove that the triangle of largest area for a given perimeter is equilateral, but I'm having some difficulties.
I've done 2 different proofs for a similar problem but for rectangles - ...
1
vote
2answers
84 views
Why is the margin of error 40 * pi * h for this problem?
For my Calculus assignment, I was given this problem:
If a right triangle has legs 6 and 8, its hypotenuse is 10. The triangle will be inscribed within a circle with area 25pi. (The hypotenuse ...
1
vote
2answers
98 views
Calculating 2D positions from a curve?
I am with quite a dylema here, as I need this for a game (so I am going to transform the given answers into programming code) to make a polygon around a curved line. from each line segment I have the ...
2
votes
1answer
136 views
Finding the minimum value of a function in an ellipse
so I have this problem for my homework:
Consider the elipse: $\dfrac {x^2}{a^2} + \dfrac{y^2}{b^2}=1$ where $0<b<a$. For every point (x,y) on the ellipse find the the perpendicular line to the ...
1
vote
2answers
191 views
What is the maximum volume of a cylinder that can fit in a sphere of a constant radius?
The first question that comes into my mind here is whether any cylinder that touches(at 4 pts) the circumference of the sphere and does not go out of it, has equal volume?
Second, how do i ...
1
vote
0answers
42 views
Planar quadrangle $ABCD$ satisfy $AB = BC$ and $CD = DA$. Compute a point $P$ such that $PA+PB+PC+PD$ is minimum
Let a planar quadrangle $ABCD$ satisfy $AB = BC$ and $CD = DA$. Compute a point $P$ such that $PA+PB+PC+PD$ is minimum, for each of following two cases.
The inner angle of $B$ is greater than $\pi$ ...
4
votes
0answers
73 views
How many points does one need for an epsilon-net
Does anyone know, how many points does one need to have an $\varepsilon$-net on a unit sphere sitting in the three-dimensional Euclidean space? Thanks!
1
vote
4answers
247 views
Geometry Prove - two perpendicular lines in a circle
In a circle of radius r, two lines (AB and CD) are perpendicular to each other and meet at X.
Show that:
1
vote
4answers
227 views
Equation of a circle that touches a line and both x and y-axes
As shown in the graph below, a circle touches the $x$-axis, the $y$-axis and a line that has equation $y = x/2 +2$.
How to find the equation of the circle?
Thanks very much!
